Here, Br is the radial component (outward positive) of the field, Bt is the coelevation component (South positive), and B# is the a/imuthal component (East positive). (See Fig. 2-5, Section 2.3.) The magnetic field literature, however, normally refers to three components X, Y, Z, consisting of North, East, and nadir relative to an oblate Earth. These components are obtained from Eq. (H-12) by
Z(" Vertical" inward positive)=£0sin< - B,cost where t=A-S<0.26, A is the geodetic latitude, and S=9O°-0 is the declination. The correction terms in sin < are of the order of 100 nT or less [Trombka and Cain, 1974],
The geocentric inertial components used in satellite work are
Note that B is still a function of longitude, <j>, which is related to the right ascension, a, by:
where Oq is the right ascension of the Greenwich meridian or the sidereal time at Greenwich (Appendix J).
Dipole Model. Equations (H-6) through (H-14) are sufficient to generate efficient computer code. However, for analytic purposes, it is convenient to obtain a dipole model by expanding the field model to first degree (n= 1) and all orders (m- 0, 1). Eq. (H-2) then becomes
= ( gfc 3cos 0+g,'a3cos <t> sine+h}a3sin <i> sin 0 ) (H-16)
The cos0 term is just the potential due to a dipole of strength g°a3 aligned with the polar axis. (See, for example, Jackson .) Similarly, the sin0 terms are dipoles aligned with the x and y axes. Relying on the principle of linear superposition, these three terms are just the Cartesian components of the dipole component of the Earth's magnetic field. From Table H-l, we find that for 1978, gf= -30109 nT
Therefore, the total dipole strength is a3//0=a3[^+?11,+AI1'],/2 = 7.943xlO,5Wbm
The coelevation of the dipole is
The East longitude of the dipole is
Thus, the first-order terrestrial magnetic field is due to a dipole with northern magnetization pointed toward the southern hemisphere such that the northern end of any dipole free to rotate in the field points roughly toward the north celestial pole. The end of the Earth's dipole in the northern hemisphere is at 78.6° N, 289.3° E and is customarily referred to as the "North" magnetic pole. Frequently, dipole models in the literature use the coordinates of the North magnetic pole and compensate with a minus sign in the dipole equation.
The above calculations were performed for 1978 by adding the secular terms to the Gaussian coefficients of epoch 1975. The location of the dipole in 1975 can be similarly calculated and compared with the 1980 dipole. That comparison yields a 0.45% decrease in dipole strength between 1975 and 1980 and a 0.071-deg drift northward and a 0.056-deg (arc) drift westward for a total motion of 0.09-deg arc.
The dipole field in local tangent coordinates is given by
The field could be converted to geocentric inertial coordinates using Eq. (H-14), but the exercise is arduous and not particularly instructive. However, we may take advantage of the dipole nature of the dominant term in the field model to approximate the magnetic field of the Earth as due to a vector dipole, m, whose magnitude and direction are given by Eqs. (H-18) through (H-20). Thus, where R is the position vector of the point at which the field is desired. Because this is a vector equation, the components of B may be evaluated in any convenient coordinate system. As an example, the field in geocentric inertial components can be obtained from the dipole unit vector, .
Bt<=2 g?cos0 + (gjcos$+A,'sin$)sin0] B9=( ^ )3 [ glsin 0 - ( g jcos$+Ajsin<#>)cos0]
sin^ cosam m= sin9'm sinam cos 8'm
where ac0 is the right ascension of the Greenwich meridian* at some reference time ("go = 98.8279° at 0h UT, December 31, 1979), dac/d/ is the average rotation rate of the Earth (360.9856469 deg/day), / is the time since reference, and = (168.6°, 109.3°) in 1978. Then m • R = /?_, sin O^cos am + /?vsin(?;sinam + R, cos$^, (H-24)
where Rx, Ry, and Rz are the geocentric inertial direction cosines of R. The field components are a3H0r - ,
These equations are useful for analytic computations and for checking computer calculations. For example, if R is in the Earth's equatorial plane, then R, =0 and a3H0
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