## Modeling The Space Environment

5.1 The Earth's Magnetic Field

5.2 The Earth's Gravitational Field

5.3 Solar Radiation and The Solar Wind Solar Radiation, The Solar Wind

5.4 Modeling the Position of the Spacecraft

5.5 Modeling the Positions of the Sun, Moon, and Planets

5.6 Modeling Stellar Positions and Characteristics

Star Catalog Data Required for Attitude Determination, Existing Star Catalogs, Generating a Core Catalog

Chapter 4 described models of the appearance, shape, and atmosphere of the Earth. This chapter is concerned with modeling properties of the spacecraft environment that are relevant to attitude determination and control. Sections 5.1 and 5.2 describe the magnetic and gravitational fields of the Earth, although many of the modeling procedures can be extended to other planets as well. Section 5.3 discusses the interplanetary medium known as the solar wind. The remaining three sections discuss models of the position of various objects needed for attitude determination—the spacecraft itself, the Sun, the Moon, the planets, and the stars.

5.1 The Earth's Magnetic Field

### Michael Plett

Although the general characteristics of the Earth's magnetic field have been known for centuries, the first systematic study of the field was initiated by the German mathematician and physicist Karl Gauss* in the early part of the nineteenth century. Since that time, a great deal of data has been accumulated, much of it as a result of spacecraft measurements during the 1960s. Although this body of data has served to increase our ability to accurately describe the field, it has not yet provided the key to the physical processes which produce it or perturb it. Thus, in this section we will describe the observed phenomena and, wherever possible, provide plausible arguments for their existence.

The Main Field. The Earth's magnetic field is predominantly that of a magnetic dipole such as that produced by a sphere of uniform magnetization or a current loop. The strength of the dipole was 7.96 X1015 Wb-m in 1975. The "south" end of the dipole was in the northern hemisphere at 78.60° N latitude and 289.55° E longitude and drifting westward at about 0.014 deg/year. The dipole strength is decreasing by 0.05%/year. This secular drift implies a possible field reversal in several thousand years. There is ambiguous evidence of several reversals

' Among his many contributions, Gauss was also the first to apply least-squares analysis to the problem of orbit determination.

in the past with time scales of 70,000 to 100,000 years between reversals [Haymes, 1971].

The plane perpendicular to the Earth-centered dipole is called the magnetic equator. The field is weakest there, being about 3 x 104 nT at the surface of the Earth. Figure 5-1 shows the variation in the dipole field strength as a function of altitude at the magnetic equator. The field strength increases by a factor of two as the magnetic latitude increases from 0 deg to 90 deg, as shown in Fig. 5-2. At the geomagnetic equator, the field is horizontal relative to the Earth's surface. At a geomagnetic latitude of about 27 deg, the field is 45 deg down from horizontal.

Fig. 5-1. Earth's Magnetic Field Intensity at the Magnetic Equator as a Function of Altitude (Adapted from Schalkowsky and Harris [1969Q

MAGNETIC LATITUDE (DEO)

Fig. 5-2. Relative Intensity of the Earth's Magnetic Field as a Function of Magnetic Latitude (Adapted from Schalkowsky and Harris [19690

MAGNETIC LATITUDE (DEO)

Fig. 5-2. Relative Intensity of the Earth's Magnetic Field as a Function of Magnetic Latitude (Adapted from Schalkowsky and Harris [19690

plots of the field strength for various altitudes are given in Figs. 5-3 and 5-4. Note that as the altitude increases, the contours become more regular and begin to resemble a dipole field more closely.

The low in magnetic intensity at about 25°S, 45°W (called the Brazilian Anomaly) together with the high at about 10°N, I00°E implies that the center of the magnetic dipole is offset from the Earth's center. In 1975, the eccentric dipole was offset 474.2 km in the direction of 19.5°N, 146.9°E [J. Bartels, 1936], The eccentric dipole is moving outward at 2.4 km/year, westward at 0.19 deg/year and northward at 0.23 deg/year. The eccentric nature of the dipole can be described mathematically as a quadrupole distribution of magnetization. The maximum deviations of the centered dipole model and the quadrupole model from the actual field of the Earth are shown in Fig. 5-5.

The fáct that the field rotates with the Earth is a clear indication that the field originates within the Earth. A coherent dipole field of this nature can be produced either by a uniformly magnetized sphere or by a current loop. However, calculations of the magnetization required lead to values much higher than those observed in the Earth's crust. Magnetization deeper than the crust is unlikely because the Curie point (i.e., the temperature at which a magnetized material loses its magnetization) of iron is reached only 20 km below the Earth's surface [Haymes, 1971],

An alternative theory postulates a dynamo effect in the outer core of the Earth driven by thermal convection currents [Garland, 1971], Basically, a dynamo is a conductor driven in a magnetic field such that it acts to sustain that field. The theory has been refined to include a primary current which produces the dipole,

Fig. 5-3. Total Magnetic Field Intensity at the Earth's Surface (in fiT Epoch 1965) (From Harris and Lyle [1969])

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