|
Summary of Kepler's Laws:
(1) The planets move in elliptical orbits, with the Sun at one focus.
(2) (called the Law of Areas) As a planet moves around its orbit, the radius vector from the Sun to the planet
sweeps out equal areas in equal times.
(3) (called the Harmonic Law) The square of the orbital period of a planet
is proportional to the cube of the semi-major axis of its orbit (P-squared = a-cubed).
The differences between Kepler's original Laws and Newton's reformulation:
Kepler didn't realize that the orbits could be any conic section, because
open conic sections (parabolas and hyperbolas) wouldn't allow the planets to keep going around the Sun. Also, because he didn't
know that the force involved was the force of gravity, he had no way of realizing, as Newton did, that the masses of the objects
would affect the results, leading to a version of the 3rd Law which includes the masses, and allows measurements of planetary
motions, satellite motions, stellar motions, or such, to be used to estimate the masses of the Sun, the planets, stars, and
galaxies.
KEPLER’S LAWS OF PLANETARY MOTION
In the sixteenth century, most people believed in the ideas of the ancient astronomer Ptolemy, that the planets,
Moon, and Sun all orbited around the Earth. Then in 1543, Nicolaus Copernicus proposed the idea that the planets and the Earth
orbited around the Sun. However, Copernicus' new theory was no better at predicting the positions of the planets in the sky
than the older, Earth-centered theory. There was still something missing.....
Half a century later, Johannes Kepler sought to refine the Copernican system and truly understand how
the planets move around the Sun. He studied observations of Mars recorded by his mentor, Tycho Brahe. Rather than trying to
force the data to support a pre-determined view of the Universe, Kepler used Tycho's observations to guide the creation of
his theories. This was a radical departure from the thought processes of his era, and it is a signal of the beginning of our
modern scientific age.
In 1609, Kepler published his first and second laws of planetary motion, The Law of Ellipses and The Equal-Areas
Law. Ten years later he published a third law, The Harmonic Law. He had succeeded in using a scientific method to create
a simple, elegant, and accurate model to describe the motion of planets around the Sun,
Kepler's First Law: The Law
of Ellipses
Previous theories of the Solar System, including those of Ptolemy and Copernicus, believed
that the orbits of the planets were perfect circles. Kepler was unable, however, to fit Tycho's observations with circular
orbits. He rejected the ancient idea of circular orbits had discovered that:
The orbits of the planets are ellipses with the Sun at one of the foci.
This is now called Kepler's First Law or The Law of Ellipses.
What is an ellipse? Glad you asked. An ellipse is a closed, curved shape that is defined
by two foci. An ellipse is a like a flattened circle. In fact, if both of the foci of an ellipse are at the same point, an
ellipse becomes a circle! If you think about it, the relationship between an ellipse and a circle is similar to the relationship
between a rectangle and a square.
An ellipse has two axes. The long one is called the major axis, and the short one is called
the minor axis. Astronomers will often use the term "semimajor axis". That's just half the length of the major axis!
The shape of an ellipse is measured by its eccentricity. The "flatter" the ellipse,
the greate the eccentricity. A circle, for example, has an eccentricity of zero since both foci are at the center. As the
ellipse becomes flatter and flatter, the foci get farther from the center, and the eccentricity will approach, but never equal,
one.
Kepler's Second Law: The Equal-Areas
Law
In addition to determining that the orbits of the planets around the Sun were ellipses, Kepler
also noticed that their speeds varied throughout their journies Kepler noted that the planets seemed to move fastest when
they were at their closest point to the Sun (called perihelion) and slowest when they were at their farthest point
from the Sun (called aphelion). Using some rather brilliant insights of geometry, Kepler discovered that:
The line that connects the planet to the Sun sweeps out equal areas in equal
times.
This is now known as Kepler's Second Law or The Equal Areas Law.
The motion this law describes also tells us that the average distance from a planet to the
Sun is equal to the length of the semimajor axis. That's why astronomers love the term so much!
Kepler's Third Law: The Harmonic
Law
After determining his first two Laws of Planetary Motion, Kepler continued to study the orbits
of the planets. Ten years later, he discovered a relation between the time of a planet's orbit nad its distance from the Sun:
The squares of the orbital periods of the planets around the Sun are proportional
to the cubes of the orbital semimajor axes.
What does this mean? This means that if you know either how much time a planet's orbit around
the Sun takes you can easily know it's average distance from the Sun, or vice-versa! Now you will often see Kepler's Third
Law written like this:
P2=a3
Where P is the orbital period in Earth years and a is the length of the semimajor axis (average
distance from the Sun) in Astronomical Units.
|
