Communications satellite Refer to Exercise. Shown in the figure is the area served by a co...

Communications satellite Refer to Exercise. Shown in the figure is the area served by a communications satellite circling a planet of radius R at an altitude a. The portion of the planet’s surface within range of the satellite is a spherical cap of depth d and surface area A = 2πRd.

(a) Express d in terms of R and θ.

(b) Estimate the percentage of the planet’s surface that is within signal range of a single satellite in equatorial synchronous orbit.

EXERCISE

Communications satellite Shown in the left part of the figure is a communications satellite with an equatorial orbit—that is, a nearly circular orbit in the plane determined by Earth’s equator. If the satellite circles Earth at an altitude of a = 22,300 mi, its speed is the same as the rotational speed of Earth; to an observer on the equator, the satellite appears to be stationary—that is, its orbit is synchronous.

(a) Using R = 4000 mi for the radius of Earth, determine the percentage of the equator that is within signal range of such a satellite.

(b) As shown in the right part of the figure, three satellites are equally spaced in equatorial synchronous orbits. Use the value of 6 obtained in part (a) to explain why all points on the equator are within signal range of at least one of the three satellites.

EXERCISE