M408M Learning Module Pages
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Chapter 10: Parametric Equations
and Polar Coordinates
Learning module LM 10.1:
Parametrized Curves:
Learning module LM 10.2: Calculus with Parametrized Curves:
Learning module LM 10.3: Polar Coordinates:
Learning module LM 10.4: Areas and Lengths of Polar Curves:
Learning module LM 10.5: Conic Sections:
Learning module LM 10.6: Conic Sections in Polar Coordinates:
Foci and directrices
Visualizing eccentricity
Polar equations for conic sections
Astronomy
Chapter 12: Vectors and the Geometry of Space
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives
Chapter 15: Multiple Integrals


Astronomy
Astronomy
Kepler's laws say that planets follow elliptical orbits around the
sun, with the sun at a focus. In other words, the distance $r$ between
a planet and the sun is given by
$$r = \frac{ed}{1e\cos(\theta\theta_0)}.$$
Perihelion and Aphelion:
The point $\theta=\theta_0+\pi$ where the planet is closest to the sun
is called perihelion, and is at distance $\displaystyle{r =
\frac{ed}{1+e}}$. The point $\theta=\theta_0$ where the planet is
farthest from the sun is called aphelion, and is at distance
$\displaystyle{r=\frac{ed}{1e}}$. (The terms perihelion and aphelion
come from the Greek word helios, meaning sun, and are only
applied to orbits around the sun.)

Most planets have very low eccentricity, and their orbits are almost
circles. The earth's eccentricity is only 0.0167, or 1.67%, so the
distance to the sun at aphelion (around July 4) is about 3.3% farther
than at perihelion (around January 3). As a result, the earth receives
about 6.5% more sunlight in January than in July! [Note: seasons are
caused by the tilt of the earth's axis, not by the eccentricity of the
orbit.]
A few planets are a bit more eccentric. Mercury's eccentricity is
about .2056. Pluto, no longer considered a planet, has an eccentricity
of .248. At perihelion, Pluto is closer to the sun than Neptune. At
aphelion, it is 60% farther.
Comets have very high eccentricity, close to 1. They start very far
from the sun, plunge into the solar system, make a close pass, and
then zip out again. Halley's Comet has an eccentricity of .967,
meaning that at aphelion it is $\displaystyle{\frac{1+e}{1e}\approx 60}$ times
farther from the sun than at perihelion. Comet HaleBopp, the
brightest comet ever seen when it passed the sun in 1997, had an
eccentricity of .995. Its aphelion is about 400 times farther than its
perihelion.
Occasionally, a comet will get a gravitational kick from a planet as
it enters the solar system. This extra energy gives it an eccentricity
slightly greater than 1. Comet C/1980 E1 had an eccentricity of
1.057. It followed a hyperbolic trajectory out of the solar system,
and will never return.


