- 4464 - KEPLER'S ORBIT MATH - is still used today? The story of how we understand planetary motion could not be told if it were not for the work of a German mathematician named Johannes Kepler. Johannes Kepler died Nov. 15, 1630, at age 58. NASA's Kepler space telescope was named for him. The spacecraft launched March 6, 2009, and spent nine years searching for Earth-like planets orbiting other stars in our region of the Milky Way
--------------- 4464 - KEPLER'S ORBIT MATH - is still used today?
- Kepler's three laws describe how planets
orbit the Sun. They describe how (1) planets move in elliptical orbits with the
Sun as a focus, (2) a planet covers the same area of space in the same amount
of time no matter where it is in its orbit, and (3) a planet’s orbital period
is proportional to the size of its orbit.
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- The planets orbit the Sun in a
counterclockwise direction as viewed from above the Sun's north pole, and the
planets' orbits all are aligned to what astronomers call the ecliptic plane.
-
- Johannes Kepler was born on Dec. 27, 1571,
in Weil der Stadt, Württemberg.
Johannes Kepler (1571-1630) was a German astronomer best known for
determining three principles of how planets orbit the Sun, known as Kepler's
laws of planetary motion.
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- As a rather frail young man, the
exceptionally talented Kepler turned to mathematics and the study of the
heavens early on. When he was six, his mother pointed out a comet visible in
the night sky. When Kepler was nine, his father took him out one night under
the stars to observe a lunar eclipse. These events both made a vivid impression
on Kepler's youthful mind and turned him toward a life dedicated to astronomy.
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- Kepler lived and worked in Graz, Austria,
during the tumultuous early 17th century. Due to religious and political
difficulties common during that era, Kepler was banished from Graz on August 2,
1600.
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- Fortunately, he found work as an assistant
to the famous Danish astronomer Tycho Brahe in Prague. Kepler moved his family
from Graz, 300 miles across the Danube River to Tycho's home.
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- Tycho is credited with making the most
accurate astronomical observations of his time, which he accomplished without
the aid of a telescope. He had been impressed with Kepler’s studies in an
earlier meeting.
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- However, some historians think Tycho
mistrusted Kepler, fearing that his bright young intern might eclipse him as
the premier astronomer of his day. Because of this, he only let Kepler see part
of his voluminous collection of planetary data.
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- Tycho assigned Kepler the task of
understanding the orbit of the planet Mars. The movement of Mars was a
problem. It didn’t quite fit the models
as described by Greek philosopher and scientist Aristotle (384 to 322 B.C.E.)
and Egyptian astronomer Claudius Ptolemy (about 100 C.E to 170 C.E.).
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- Aristotle thought Earth was the center of
the universe, and that the Sun, Moon, planets, and stars revolved around it.
Ptolemy developed this concept into a standardized, geocentric model based around Earth as a stationary object, at
the center of the universe.
-
- Historians think that part of Tycho’s
motivation for giving the Mars problem to Kepler was Tycho's hope that it would
keep Kepler occupied while Tycho worked to perfect his own theory of the solar
system. That theory was based on a geocentric model, modified from Ptolemy's,
in which the planets Mercury, Venus, Mars, Jupiter, and Saturn all orbit the
Sun, which in turn orbits Earth.
-
- As it turned out, Kepler, unlike Tycho,
believed firmly in a model of the solar system known as the heliocentric model,
which correctly placed the Sun at its center. This is also known as the
Copernican system, because it was developed by astronomer Nicolaus Copernicus
(1473-1543). But the Copernican system incorrectly assumed the orbits of the
planets to be circular.
-
- Like many philosophers of his era, Kepler
had a mystical belief that the circle was the universe’s perfect shape, so he
also thought the planets’ orbits must be circular. For many years, he struggled
to make Tycho’s observations of the motions of Mars match up with a circular
orbit.
-
- Kepler eventually realized that the orbits
of the planets are not perfect circles. His brilliant insight was that planets
move in elongated, or flattened, circles called ellipses.
-
- The particular difficulties Tycho had with
the movement of Mars were due to the fact that its orbit was the most
elliptical of the planets for which he had extensive data. Thus, in a twist of
irony, Tycho unwittingly gave Kepler the very part of his data that would
enable his assistant to formulate the correct theory of the solar system.
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- The orbits of the planets are ellipses:
-
------------------- An ellipse is defined by two points, each
called a focus, and together called foci. The sum of the distances to the foci
from any point on the ellipse is always a constant.
\-
------------------ The amount of flattening of the ellipse is
called the eccentricity. The flatter the ellipse, the more eccentric it is.
Each ellipse has an eccentricity with a value between zero (a circle), and one
(essentially a flat line, technically called a parabola).
-
------------------- The longest axis of the ellipse is called the
major axis, while the shortest axis is called the minor axis. Half of the major
axis is termed a semi-major axis.
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- After determining that the orbits of the
planets are elliptical, Kepler formulated three laws of planetary motion, which
accurately described the motion of comets as well.
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- In 1609 Kepler published “Astronomia Nova,”
which explained what are now called Kepler's first two laws of planetary
motion. Kepler had noticed that an imaginary line drawn from a planet to the
Sun swept out an equal area of space in equal times, regardless of where the
planet was in its orbit. If you draw a triangle from the Sun to a planet’s
position at one point in time and its position at a fixed time later, the area
of that triangle is always the same, anywhere in the orbit.
-
- For all these triangles to have the same
area, the planet must move more quickly when it’s near the Sun, but more slowly
when it is farther from the Sun. This discovery became Kepler’s second law of
orbital motion, and led to the realization of what became Kepler’s first law:
that the planets move in an ellipse with the Sun at one focus point, offset
from the center.
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- In 1619, Kepler published “Harmonices
Mundi,” in which he describes his "third law." The third law shows
that there is a precise mathematical relationship between a planet’s distance
from the Sun and the amount of time it takes revolve around the Sun.
-
------------- Kepler's First Law: Each planet's orbit about
the Sun is an ellipse. The Sun's center is always located at one focus of the
ellipse. The planet follows the ellipse in its orbit, meaning that the planet-to-Sun
distance is constantly changing as the planet goes around its orbit.
-
-------------- Kepler's Second Law: The imaginary line
joining a planet and the Sun sweeps out
equal areas of space during equal time intervals as the planet orbits.
Basically, the planets do not move with constant speed along their orbits.
Instead, their speed varies so that the line joining the centers of the Sun and
the planet covers an equal area in equal amounts of time.
-
- The point of nearest approach of the planet
to the Sun is called perihelion. The point of greatest separation is aphelion,
hence by Kepler's second law, a planet is moving fastest when it is at
perihelion and slowest at aphelion.
-
---------------- Kepler's Third Law: The orbital period of a
planet, squared, is directly proportional to the semi-major axes of its orbit,
cubed. This is written in equation form as p^2=a^3. Kepler's third law implies
that the period for a planet to orbit the Sun increases rapidly with the radius
of its orbit. Mercury, the innermost planet, takes only 88 days to orbit the
Sun. Earth takes 365 days, while distant Saturn requires 10,759 days to do the
same.
-
- Kepler didn’t know about gravity, which is
responsible for holding the planets in their orbits around the Sun, when he
came up with his three laws. But Kepler’s laws were instrumental in Isaac
Newton’s development of his theory of universal gravitation, which explained
the unknown force behind Kepler's third law.
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- Kepler and his theories were crucial in the
understanding of solar system dynamics and as a springboard to newer theories
that more accurately approximate planetary orbits. However, his third law only
applies to objects in our own solar system.
-
- Newton’s version of Kepler’s third law
allows us to calculate the masses of any two objects in space if we know the
distance between them and how long they take to orbit each other (their orbital
period). What Newton realized was that the orbits of objects in space depend on
their masses, which led him to discover gravity.
-
- Newton’s generalized version of Kepler’s
third law is the basis of most measurements we can make of the masses of
distant objects in space today. These applications include determining the
masses of moons orbiting the planets, stars that orbit each other, the masses
of black holes (using nearby stars affected by their gravity), the masses of
exoplanets (planets orbiting stars other than our Sun), and the existence of
mysterious dark matter in our galaxy and others.
-
- In planning trajectories (or flight plans)
for spacecraft, and in making measurements of the masses of the moons and
planets, modern scientists often go a step beyond Newton. They account for
factors related to Albert Einstein’s theory of relativity, which is necessary
to achieve the precision required by modern science measurements and
spaceflight.
-
- However, Newton’s laws are still accurate
enough for many applications, and Kepler’s laws remain an excellent guide for
understanding how the planets move in our solar system.
-
- The Kepler space telescope left a legacy of
more than 2,600 planet discoveries from outside our solar system, many of which
could be promising places for life.
-
-
May 9, 2024 4464
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