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AZIMUTH, t FOR SUN ANGLE VARIATION COMPUTATIONS (DEG) 7B706S60&5 604S40 35 302S20

IS to

20 2S 30 3&404BS0SS60G&70 7S AZIMUTH. FOR SPIN RATE VARIATION COMPUTATIONS IOEGI

Fig. 16-9. Ratio, Rp, of Observed Sun Angle Variation to Nutation Angle as a Function of Sun Sensor Azimuth, {, and R/ lor Deg, Deg, and lx<ly<lt. Read ( from bottom for

Rp used in' computing &P and from the top in computing 9mx from the Sun angle variation. See text for explanation and example.

20 2S 30 3&404BS0SS60G&70 7S AZIMUTH. FOR SPIN RATE VARIATION COMPUTATIONS IOEGI

Fig. 16-9. Ratio, Rp, of Observed Sun Angle Variation to Nutation Angle as a Function of Sun Sensor Azimuth, {, and R/ lor Deg, Deg, and lx<ly<lt. Read ( from bottom for

Rp used in' computing &P and from the top in computing 9mx from the Sun angle variation. See text for explanation and example.

approximate analytic form

The analytic approximation is about 5% low at Rf=0.25 and about 25% low at Rg= 0.05. The oscillations of Pm with changing \f> defined by Eq. (16-118) vary from sinusoidal to nearly sawtooth depending on the spacecraft symmetry and slit location. (Note that on Fig. 16-8, points B and D are only 100 deg apart in azimuth.) The rate of the Sun angle oscillation for an asymmetric spacecraft is the i.e., asymmetric analog of Eq. (16-100) for small 0^,, where the intermediate variable, R„ is defined by

with R, positive for lx,ly < Iz and negative for lt < lx,ly

The same numerical process that yields Rfi also yields £m, the azimuth of the angular momentum vector when /3m is a maximum. This information is valuable for phasing a torque to counteract the nutation and is shown in Fig. 16-10 for the same conditions as Fig. 16-9. Note that £m is compressed toward 90 deg as the spacecraft becomes more asymmetrical (smaller Rg). These curves are also insensitive to both and /?. As long as 30^ < fi and Rf>0A5, the curves are accurate to about 15

SUN SENSOR SLIT AZIMUTH. ( IDEGI

Fig. 16-10. Azimuth, 4„. of Angular Momentum Vector, L, at Which the Measured Sun Angle is a Maximum as a Function of Slit Azimuth, and Rf for 9max = 2 Deg and /?=90 Deg. See text for explanation and example.

SUN SENSOR SLIT AZIMUTH. ( IDEGI

Fig. 16-10. Azimuth, 4„. of Angular Momentum Vector, L, at Which the Measured Sun Angle is a Maximum as a Function of Slit Azimuth, and Rf for 9max = 2 Deg and /?=90 Deg. See text for explanation and example.

deg in im. Figures 16-9 and 16-10 are intended for estimates only; Eq. (16-118) should be solved directly when spacecraft parameters are well established.

To illustrate nutation monitoring from Sun angle data, consider a spacecraft with relative moments of inertia 7^ = 51.5, A, = 71.3, lz = 90.0 and a Sun sensor mounted 30 deg from the x axis, i.e., ¿ = 30°. From Eq. (16-114), /?/ = 0.35. Assume that covers the range 67.5° <Pm< 73.5°. Rfi is obtained from Fig. 16-9 as 0.72 and is then computed from

From Fig. 16-10, the maximum value of f$m occurs at £m = 59°, which means that L is at an azimuth of 59 deg at the measurement of the maximum Sun angle. A torque applied 180° out of phase with L at an azimuth of 239 deg would reduce the nutation amplitude. This technique was used successfully on the SSS-1 spacecraft [Flatley, 1972b],

Many Sun sensors provide the Sun crossing time so that the interval between successive crossings may be used to measure the spin period. However, this measured spin period is affected by nutation. By examining Fig. 15-8 and assuming that the Sun sensor points in the direction of P3 to R on that figure, we see that the Sun sensor is rotating counterclockwise in inertial space at an approximately uniform rate (or, equivalently, the Sun is rotating clockwise relative to the sensor). As shown in Fig. 16-11, the angular momentum vector is rotating counterclockwise relative to the Sun sensor, where, for simplicity, we have chosen lx = ly = lT<lt. Because 1T is less than the spacecraft is nutating more rapidly than it is rotating and L is rotating faster than the Sun sensor.

It is essentially the wobbling motion of the spacecraft which is responsible for the variation in the measured spin period. Assume that in Fig. 16-11, L and Sare at L, and Sx at time /,. At time t2, after one measured spin period, the Sun has rotated 360 deg to S2. L has moved more rapidly and gone more than 360 deg to

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