The frequency of the storms is somewhat correlated with Sun-spot activity, since flares are usually associated with Sun spots [Haymes, 1971]. Thus, a storm may recur after 27 days (the length of a solar rotation) and the overall activity tends to follow the 11-year solar cycle. Storms are also more frequent near the equinoxes, possibly because at approximately those times, the position of the Earth is at the highest solar latitude (about 7 deg). Sun spots appear most frequently between the solar latitudes of 5 deg and 40 deg on both sides of the equator. They appear first at the higher solar latitudes at the beginning of a solar cycle.
The geomagnetic field is monitored continuously at a series of stations called magnetic observatories. They report observed magnetic activity, such as storms, as an index, K, which is the deviation of the most disturbed component of the field from the average quiet-day value [Chernosky, Fougere, and Hutchinson, 1965]. The K scale is quasi-logarithmic with tf=0, quiet, and K — 9, the largest disturbance the station is likely to see. The value of K is averaged and reported for every 3 hours. The values of K for 12 selected stations are corrected for the station's geomagnetic latitude (since activity is latitude dependent) and then averaged to produce the planetary index, Kp. The indices are published each month in the Journal of Geophysical Research. The value of Kp is a good indicator of the level of magnetic storms and is therefore an indication of the deviation of the geomagnetic field from the model in Eq. (5-5). The 3-hourly planetary index can be roughly (10%) converted to the linear 3-hourly planetary amplitude, ap, by:
The value of ap is scaled such that at 50 deg geomagnetic latitude and a deviation of 500 nT at K=9, the field deviation A£ = |£dislurbed- £qujet| is
For other latitudes, ap is scaled by dividing the lower limit of hB for K=9 by 250. Thus, at a higher latitude for which AB= 1000 nT corresponds to A"=9, bB»4ap (5-8)
Although Kp is a measure of geomagnetic activity, it is ultimately a measure of solar activity. In fact, it has been found empirically that the velocity of the solar wind can be derived from Kp by:
where v is in kilometres per second and "S,Kp is the sum over the eight values of Kp for the day [Haymes, 1971], Similarly, the interplanetary magnetic field is generated by the Sun, and it has been shown empirically that the interplanetary field is approximately
where B is the magnitude of the interplanetary field in nT [Haymes, 1971].
The Sun is also responsible for the diurnal variation of the geomagnetic field. Solar electromagnetic radiation ionizes some atmospheric atoms and molecules at an altitude of roughly 100 km, producing the E-layer of the ionosphere. The Sun'i gravitational field then exerts a tidal force causing the ions and electrons to rise.
The interaction of the charged particles with the geomagnetic field produces a rather complex current system which creates a magnetic field. The effect is most pronounced on the day side of the Earth, since it is dependent on the ion density of the E-layer. On solar quiet days, this field causes a deviation from the internal field of 20-40 nT in the middle latitude regions and can cause deviations of 100-200 nT near the magnetic equator [Harris and Lyle, 1969]. At each magnetic observatory, the daily magnetic variations for the five quietest days are averaged together to produce the quiet day solar variation, S . This variation is subtracted from the actual variations before generating Kp. The Moon also exerts daily tidal forces which lead to quiet-day variations about 1 /30 of that due to the Sun [Harris and Lyle, 1969].
There are two other current systems of some importance: the polar electrojet and the equatorial electrojet. The polar electrojet is an intense ionospheric current that flows westward at an altitude about 100 km in the auroral zone. Changes in the electrojet can cause negative excursions (called bays) as great as 2000 nT and are typically about 1000 to 1500 nT at the Earth's surface. The excursions can last from 0.5 to 2 hours. Like magnetic storms, auroral activity has a 27-day periodicity and reaches a maximum at the equinox« [Harris and Lyle, 1969]. The equatorial electrojet is an intense west-to-east current in the sunlit ionosphere. It is partly responsible for the high intensity of the magnetic storms. It produces a 220 nT discontinuity in the total field between 96- and 130-km altitude. At 400 km, the field is 30 to 40 nT at longitudes across South America and 10 to 20 nT elsewhere [Zmuda, 1973].
52 The Earth's Gravitational Field
Two point masses, M and m, separated by a vector distance r, attract each other with a force given by Newton's law of gravitation as r where G is the gravitational constant (see Appendix M). If A/© is the mass of the Earth and m is the mass of the body whose motion we wish to follow, then it is convenient to define the geocentric gravitational constant, p9, and the Earth gravitational potential, U, by:
From Eqs. (5-12) and (5-13), Eq. (5-11) may be rewritten as the gradient of a scalar potential:
where r is the unit vector from the Earth's center to the body (assumed to be a point mass). A gravitational potential satisfying Eq. (5-14) may always be found due to the conservative nature of the gravitational field.
By extending the single point mass, m, to a collection of point masses, the gravitational potential at a point outside a continuous mass distribution over a finite volume can be defined. For example, consider a solid body of density p, situated in a rectangular coordinate system with the mass elements coordinates denoted by (|, if, f) and the point coordinates denoted by (x,y,z). The gravitational potential at the point, P=(x,y,z), due to the body can be written as [Battin, 1963]
Successive applications of Gauss' law and the divergence theorem show that U satisfies Poisson's equation,
which, in the region exterior to the body (i.e., where p = 0), reduces to Laplace's equation:
Because of the spherical symmetry of most astronomical objects, it is convenient to write Eq. (5-15) in the spherical coordinate system (r, 9, <f>). In this case, solutions to Eq. (5-17) may be written in terms of spherical harmonics as described in Appendix G. Specifically, U for the Earth can be expressed in the convenient form
where B(r,0,<j>) is the appropriate spherical harmonic expansion to correct the gravitational potential for the Earth's nonsymmetric mass distribution. B(r,$,<j>) may be written explicitly* as [Meirovitch, 1970; Escobal, 1965]
Here, R& is the radius of the Earth, Jn are zonal harmonic coefficients, Pn m are Legendre polynomials, and Cnm and are tesseral harmonic coefficients for nj>m and sectoral harmonic coefficients for n = m (see Appendix G).
In Eq. (5-19), we see that the zonal harmonics depend only on latitude, not on longitude. These terms are a consequence of the Earth's oblateness. The tesseral
'Note that the n=0 term is written explicitly in Eq. (5-18) as--—, and that the n = 1 term is absent due to the origin of the coordinate system being coincident with the Earth's center of mass.
harmonics represent longitudinal variations in the Earth's shape. Although generally smaller than zonal terms, tesseral components become important in the case of geosynchronous spacecraft because the satellite remains nearly fixed relative to the Earth; consequently, longitudinal variations do not average to zero over a long period of time. For most satellites other than geosynchronous ones, the assumption of axial symmetry of the Earth is usually valid, and only the zonal harmonic corrections are needed. Thus, the expression for the gravitational potential of the Earth can be approximated as
The zonal harmonics are a major cause of perturbations for Earth-orbiting spacecraft, being the primary source of changes in orbital period, longitude of the ascending node, and argument of perigee (Section 3.4).
The gravitational potential of Eq. (5-18), when combined with the potential due to the angular momentum of the Earth's rotation, describes a mathematical model or reference figure for the shape of the Earth, known as the geoid or mean sea level. The geoid is a surface coincident with the average sea level (i.e., less meteorological and tidal effects) over the globe or in an imaginary channel cut in the continents. A number of measurement techniques have been used [King-Hele, 1976] to map the geoid, including satellite-to-satellite tracking, in which a geosynchronous satellite measures the relative velocity of a lower orbiting satellite; radar altimetry from satellites; and laser ranging to reflectors both on satellites and the Moon. The last method has been the most accurate.
Figure 5-8 illustrates the geoid height or deviation of the geoid from a reference n r
spheroid of flattening 1/298.255 and semimajor axis of 6378.145 km* as given by the Goddard Earth Model-8 (GEM-8) [Wagner, et al., 1976]. Particularly noticeable is the variation from 77 m above the geoid near New Guinea to 105 m below in the Indian Ocean.
The accuracy of the potential function, i.e., the number of terms included in the infinite spherical harmonic series, has a greater effect on orbital dynamics than on attitude dynamics. For analysis of gravity-gradient torques on spacecraft, inclusion of the J2 term in the harmonic series is normally sufficient because of the uncertainties in other environmental disturbance torques. For practical purposes, the point mass potential function, Eq. (5-14), is adequate for spinning satellites or those with only short appendages. Table 5-4 lists the differential acceleration, da/dr = -2ft/r\ experienced by these satellites for various altitudes.
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