(Solved): A convenient way to describe the size of a celestial object is to measure the angle \( \theta \) t ...

A convenient way to describe the size of a celestial object is to measure the angle \( \theta \) that this object subtends when observed from the earth or spacecraft. Optical sensor design often requires knowledge of the angle that the object of observation subtends when observed from a spacecraft, for example, to study an asteroid by an imaging system on a flyby mission. In space communications, the angles subtended by objects are needed to design antennas, for example, to cover the hemisphere from geostationary or GPS orbit. a. Derive the analytical formula to calculate the angle subtended by a spherical object of the radius \( R_{0} \) by an observer at the distance \( L \) from the center of the object. b. Simplify (using the Taylor series expansion) the formula for the case of small angles \( \theta \). c. The altitude of the GPS orbit is \( h_{G P S}=10898.0 \) n.mi. (n.mi. = nautical mile). Assume the Earth to be a sphere with a radius \( R_{E}=6378.14 \mathrm{~km} \). Calculate the angular size of (the plane angle, \( \theta \), subtended by) the Earth as viewed from a GPS spacecraft and the error of using the approximate formula (obtained in b).