## R

which is positive, i.e., north. Because the direction of the field line is customarily defined as that indicated by a compass needle, Eq. (H-22) is self-consistent.

For analytical work, the most useful coordinate system is the 1. b. n orbit plane system (Section 2.2), in which R has the particularly simple representation

where v' is the true anomaly measured from the ascending node. Vectors are transformed into the l,b,n system from the geocentric inertial system by first rotating about the inertial z axis through fl, the right ascension of the ascending node, followed by a rotation about the ascending node by the angle /'. the orbital inclination. Using this transformation, the unit magnetic dipole is m, = sin 0^cos (S2 - amj mb= - sin 0^cos/'sin(S2-<*„,) +cos 0^sin/ (H-28)

* This technique of computing ac is good to about 0.003° for I year on either side of the reference date. At times more" distant from the reference date, a new aco can be computed as described in Appendix J. Note that Oq,, is equal to the Greenwich sidereal time at the reference time of 0h UT. December 31. 1978.

where J2 is the right ascension of the ascending node and i is the inclination of the orbit.

Substituting Eqs. (H-27) and (H-28) into Eq. (H-22) yields the magnetic field in the l,b,n system. Although the equations are moderately complex, they can still be useful. Due to the simple form for R, especially for circular orbits, it is possible to analytically integrate the torque due to a spacecraft dipole moment as has been done for ITOS [Kikkawa, 1971].

A circular equatorial orbit is particularly simple because i'=fl=0 and, therefore, m • R = sin (cos omcos v' + sin amsin v') (H-29)

Substituting into Eq. (H-22) and simplifying yields a3H0

As in Eq. (H-26), the minus sign in the orbit normal component, B„ assures the northward direction of the field lines.

The torque resulting from a spacecraft magnetic dipole interacting with Bn is in the orbit plane, or, in this case, the equatorial plane. This torque causes precession around the orbit normal, or, for /=0, right ascension motion. Torque out of the orbit plane is caused by the ascending node component B, and the component Bb. For / = 0, out of plane is the same as declination motion. Thus, for an equatorial orbit, the ratio of declination motion to right ascension motion is at most on the order of

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