James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

686 Chapter 10 Parametric Equations and Polar Coordinates
In Figure 6 we use a computer to sketch a number of conics to demonstrate the effect
of varying the eccentricity e. Notice that when e is close to 0 the ellipse is nearly circular,
whereas it becomes more elongated as e l 1 2 . When e − 1, of course, the conic is a
parabola.
e=0.1
e=0.5
e=0.68
e=0.86
e=0.96
FIGURE 6
e=1 e=1.1 e=1.4 e=4
Kepler’s Laws
In 1609 the German mathematician and astronomer Johannes Kepler, on the basis of huge
amounts of astronomical data, published the following three laws of planetary motion.
Kepler’s Laws
1. A planet revolves around the sun in an elliptical orbit with the sun at one focus.
2. The line joining the sun to a planet sweeps out equal areas in equal times.
3. The square of the period of revolution of a planet is proportional to the cube of
the length of the major axis of its orbit.
Although Kepler formulated his laws in terms of the motion of planets around the sun,
they apply equally well to the motion of moons, comets, satellites, and other bodies that
orbit subject to a single gravitational force. In Section 13.4 we will show how to deduce
Kepler’s Laws from Newton’s Laws. Here we use Kepler’s First Law, together with the
polar equation of an ellipse, to calculate quantities of interest in astronomy.
For purposes of astronomical calculations, it’s useful to express the equation of an
ellipse in terms of its eccentricity e and its semimajor axis a. We can write the distance d
from the focus to the directrix in terms of a if we use (4):
a 2 − e2 d 2
s1 2 e 2 d ? d 2 − a 2 s1 2 e 2 d 2
? d − as1 2 e2 d
2 e 2 e
So ed − as1 2 e 2 d. If the directrix is x − d, then the polar equation is
r −
ed
1 1 e cos − as1 2 e2 d
1 1 e cos
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