Calculus: Early Transcendental Functions, 4th Edition (Smith)

Chapter 3: Applications of Differentiation

 (24.0K) The Solar and Heliospheric Observatory (SOHO) is an international project for the observation and exploration of the Sun. The National Aeronautics and Space Administration (NASA)is responsible for operations of the SOHO spacecraft, including periodic adjustments to the spacecraft's location to maintain its position directly between the Earth and the Sun. With an uninterrupted view of the sun, SOHO can collect data to study the internal structure of the Sun, its outer atmosphere and the solar wind. SOHO has produced numerous unique and important images of the Sun, including the discovery of acoustic solar waves moving through the interior and false color images showing the velocity patterns on the surface of the Sun. SOHO is in orbit around the Sun, located at a relative position called the L1 Lagrange point for the Sun-Earth system. This is one of five points at which the gravitational pulls of the Sun and the Earth combine to maintain a satellite's relative position to the Sun and Earth. In the case of the L1 point, that position is on a line between the Sun and the Earth, giving the SOHO spacecraft (see above) a direct view of the Sun and a direct line of communication back to the Earth. Because gravity causes the L1 point to rotate in step with the Sun and Earth, little fuel is needed to keep the SOHO spacecraft in the proper location. Lagrange points are solutions of "three-body" problems, in which there are three objects with vastly different masses. The Sun, the Earth and a spacecraft comprise one example, but other systems also have significance for space exploration. The Earth, the Moon and a space lab is another system of interest; the Sun, (16.0K) Jupiter and an asteroid is a third system. The clusters of asteroids at the L4 and L5 Lagrange points of the Sun-Jupiter system are called Trojan asteroids. For a given system, the locations of the five Lagrange points can be determined by solving equations. As you will see in the section 3.1 exercises, the equation for the location of SOHO is a difficult fifth-order polynomial equation. For a fifth-order equation, we usually are forced to gather graphical and numerical evidence to approximate solutions. The graphing and analysis of complicated functions and the solution of equations involving these functions are the emphases of this chapter.