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Applications of Differentiation
CHAPTER
The Solar and Heliospheric Observatory (SOHO) is an inter-
national 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 atmo- 3
sphere and the solar wind. SOHO has produced numerous
unique images of the Sun, including the discovery of acoustic
solar waves moving through the interior.
SOHO is in orbit around the Sun, located at a relative
position called the L Lagrange point for the Sun-Earth sys-
1
tem. 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 L 1
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
step with the Sun and Earth, little fuel is needed to keep the SOHO spacecraft in the
line of communication back to the Earth. Because gravity causes the L point
proper location. 1
to rotate in step with the Sun and Earth, little fuel is needed to keep the SOHO
Lagrange points are solutions of “three-body” problems, in which there are three
spacecraft in the proper location.
objects with vastly different masses. The Sun, the Earth and a spacecraft comprise one
Lagrange points are solutions of “three-body” problems, in which there
example, but other systems also have significance for space exploration. The Earth, the
are three objects with vastly different masses. The Sun, the Earth and a space-
Moon and a space lab is another system of interest; the Sun, Jupiter and an asteroid is a
craft comprise one example, but other systems also have significance for space
third system. There are clusters of asteroids (called Trojan asteroids) located at the L
4
exploration. The Earth, the Moon and a space lab is another system of inter-
and L Lagrange points of the Sun-Jupiter system.
5
est; the Sun, Jupiter and an asteroid is a third system. There are clusters of
Fora given system, the locations of the five Lagrange points can be determined by
asteroids (called Trojan asteroids) located at the L and L Lagrange points of
4
5
solving equations. As you will see in the section 4.1 exercises, the equation for the loca-
the Sun-Jupiter system.
tion of SOHO is a difficult fifth-order polynomial equation. For a fifth-order equation,
For a given system, the locations of the five Lagrange points can be deter-
we usually are forced to gather graphical and numerical evidence to approximate solu-
mined by solving equations. As you will see in the section 3.1 exercises, the
tions. The graphing and analysis of complicated functions and the solution of equations
equation for the location of SOHO is a difficult fifth-order polynomial equa-
involving these functions are the emphases of this chapter.
tion. For a fifth-order equation, we usually are forced to gather graphical and
(t)Ton Kinsbergen/Science
Source, (b)Christian Jegou
Publiphoto Diffusion/
Science Source Wave inside the Sun L orbit
1
211
Christian Jegou Publiphoto Diffusion/Science Source

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