Who said a planet moves faster in the part of its orbit nearer the Sun and slower when farther from the Sun sweeping out equal areas in equal times?

Kepler’s Laws of Planetary Motion — The astronomer Johannes Kepler’s main contribution to astronomy was his three laws of planetary motion. Kepler found these laws empirically by studying extensive observations recorded by Tycho Brahe. He found the first two laws in 1609 and the third one in 1618. Isaac Newton was later able to derive the laws from his laws of motion and gravity, thereby producing strong evidence in favor of Newton’s inverse-square gravitational law. Kepler’s First Law (1609): The orbit of a planet about the Sun is an ellipse with the Sun at one focus. Kepler’s Second Law (1609): A line joining a planet and the Sun sweeps out equal areas in equal intervals of times. Kepler’s Third Law (1618): The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit. Kepler’s First Law There is no object at the other focus of a planet’s orbit. The semimajor axis, a, of an orbit is the average distance between the planet and the Sun. As a planet travels in its elliptical orbit, its distance, from the Sun, and speed vary. A planet moves most rapidly when it is nearest the Sun, or at perihelion. A planet moves most slowly when it is farthest from the Sun, or at aphelion. Kepler’s Second Law This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from point A to B. During this day, an imaginary line from the Sun to the planet will sweep out an area. The same area will be swept every day. Kepler’s Third Law (Harmonic Law) – P = object’s sidereal period in years – a = object’s semimajor axis, in AU – P2 = a3 The larger the distance from the Planet to the Sun, a, the longer the sidereal period. From this one can show that the larger the orbit is, the slower the average speed of the orbiting object will be. Not Just Applicable to Planets The laws are applicable whenever a comparatively light object revolves around a much heavier one because of gravitational attraction. It is assumed that the gravitational effect of the lighter object on the heavier one is negligible. An example is the case of a satellite revolving around Earth. Kepler’s Understanding of Said Laws Kepler did not understand why his laws were correct, it was Isaac Newton who discovered the answer to this. —–

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Planetary Physics

Kepler's three laws describe how planetary bodies 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 (its semi-major axis).

Explore the process that Johannes Kepler undertook when he formulated his three laws of planetary motion.

Transcript

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.

The story of our greater understanding of planetary motion could not be told if it were not for the work of a German mathematician named Johannes Kepler. Kepler lived 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 2nd, 1600.

Fortunately, an opportunity to work as an assistant for the famous astronomer Tycho Brahe presented itself and the young Kepler moved his family from Graz 300 miles across the Danube River to Brahe's home in Prague. Tycho Brahe is credited with the most accurate astronomical observations of his time and was impressed with the studies of Kepler during an earlier meeting. However, Brahe mistrusted Kepler, fearing that his bright young intern might eclipse him as the premier astronomer of his day. He, therefore, led Kepler to see only part of his voluminous planetary data.

He set Kepler, the task of understanding the orbit of the planet Mars, the movement of which fit problematically into the universe as described by Aristotle and Ptolemy. It is believed that part of the motivation for giving the Mars problem to Kepler was Brahe's hope that its difficulty would occupy Kepler while Brahe worked to perfect his own theory of the solar system, which was based on a geocentric model, where the earth is the center of the solar system. Based on this model, the planets Mercury, Venus, Mars, Jupiter, and Saturn all orbit the Sun, which in turn orbits the earth. As it turned out, Kepler, unlike Brahe, believed firmly in the Copernican model of the solar system known as heliocentric, which correctly placed the Sun at its center. But the reason Mars' orbit was problematic was because the Copernican system incorrectly assumed the orbits of the planets to be circular.

After much struggling, Kepler was forced to an eventual realization that the orbits of the planets are not circles, but were instead the elongated or flattened circles that geometers call ellipses, and the particular difficulties Brahe hand with the movement of Mars were due to the fact that its orbit was the most elliptical of the planets for which Brahe had extensive data. Thus, in a twist of irony, Brahe unwittingly gave Kepler the very part of his data that would enable Kepler to formulate the correct theory of the solar system, banishing Brahe's own theory.

Since the orbits of the planets are ellipses, let us review three basic properties of ellipses. The first property of an ellipse: 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 second property of an ellipse: 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 third property of an ellipse: 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. Knowing then that the orbits of the planets are elliptical, johannes Kepler formulated three laws of planetary motion, which accurately described the motion of comets as well.

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 orbital ellipse. The Sun is at one focus. 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 equal areas of space during equal time intervals as the planet orbits. Basically, that planets do not move with constant speed along their orbits. Rather, their speed varies so that the line joining the centers of the Sun and the planet sweeps out equal parts of an area in equal times. The point of nearest approach of the planet to the Sun is termed 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 squares of the orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun. The earth takes 365 days, while Saturn requires 10,759 days to do the same. Though Kepler hadn't known about gravitation when he came up with his three laws, they were instrumental in Isaac Newton deriving his theory of universal gravitation, which explains the unknown force behind Kepler's Third Law. Kepler and his theories were crucial in the better understanding of our solar system dynamics and as a springboard to newer theories that more accurately approximate our planetary orbits.

Three laws devised by Johannes Kepler to define the mechanics of planetary motion. The first law states that planets move in an elliptical orbit, with the Sun being one focus of the ellipse. This law identifies that the distance between the Sun and Earth is constantly changing as the Earth goes around its orbit. The second law states that the radius of the vector joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. As such, the planet moves quickest when the vector radius is shortest (closest to the Sun), and moves more slowly when the radius vector is long (furthest from the Sun). The third law states that the ratio of the squares of the orbital period for two planets is equal to the ratio of the cubes of their mean orbit radius. This indicates that the length of time for a planet to orbit the Sun increases rapidly with the increase of the radius of the planet's orbit.

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