Fig. 4-16. Meridian Cross Section of the Earth Showing the Horizon Spheroid and the Horizon Bane.

(In this Figure, the observer is in yz plane.)

The normal to the horizon plane depends only on the angular position of the observer. Thus, as the observer moves along a fixed nadir* direction, he sees a set of parallel horizon planes. As shown in Fig. 4-17, when the observer is on the Earth's surface (point A), the horizon plane is just the tangent plane at that point. As the observer moves to a distance, d, from the center of the Earth (point B), the parallel horizon plane will intersect the nadir line at a distance

* Nadir hi re means toward the Earth's center.

from the Earth's center, where R is the distance from the Earth's center to point A . (the subobserver or subsatellite point). R is given by.

The horizon plane will approach the center of the Earth as the observer approaches infinity. Note that the nadir line passes through the center of the horizon ellipse.

To find the shape of the horizon ellipse or, equivalently, the shape of the Earth as seen by the observer, it is convenient to solve Eqs. (4-16) and (4-21) in the local tangent coordinate system defined by N, E, and Z through P, as shown in Fig. 3-6. It can be shown that the angular radius of the Earth or the horizon of the Earth is given by p=arc cot

where X is the geocentric latitude of the observer's position and d and R are the distances from the center of the Earth to the observer and the subobserver point, respectively. As shown in Fig. 4-18, * is the azimuth angle of the horizon vector,

H, in local tangent coordinates and p is the angle between the nadir vector and the horizon vector. When /= 0, that is, when the Earth is spherical, Eq. (4-24) reduces to p=arc sin (a/d) (4-25)

as expected.

Figure 4-18 shows an example of the shape of the Earth as seen by an observer at 45-deg geocentric latitude and a distance of 200 km above the Earth's surface. To make the oblateness effect noticeable, a flattening factor 100 times larger than the true value was used. Table 4-3 compares the angular radius of the Earth for spheroidal and spherical Earth models.

Table 4-3. Angular Radius of the Earth From the Spheroidal Model at an Altitude of 200 km and Geocentric Latitude of 45 deg. Equation (4-24) and the following parameters were used: 0 = 6,378.14 km,/=>0.00335281, d=a + 200 km. The angular radius of a spherical Earth of radius a is p0 = 75.8353 deg.

Table 4-3. Angular Radius of the Earth From the Spheroidal Model at an Altitude of 200 km and Geocentric Latitude of 45 deg. Equation (4-24) and the following parameters were used: 0 = 6,378.14 km,/=>0.00335281, d=a + 200 km. The angular radius of a spherical Earth of radius a is p0 = 75.8353 deg.

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