What is the relationship between the orbital distance of a planet from the Sun and its period of orbit?

There are 8 planets (and one dwarf planet) in orbit around the sun, hurtling around at tens of thousands or even hundreds of thousands of miles an hour. Because the planets are attracted to the sun and kept in orbit by its gravitational force, the distance they are away from the sun is directly related to the speed at which they travel.

Explore the relationship between orbital period distance.

What will have a larger orbital period, short strings or long strings?

• Twine
• Metal Washer
• Scissors
• Meter stick
• Stopwatch
• Notebook and pen or pencil

1. Cut several different lengths of twine and measure them. Record the measurements in your notebook.
2. Tie a washer securely to one end of the twine.
3. Start swinging the washer around your head so it barely keeps tension on the rope.
4. Have a partner start the stopwatch and at the same time, start counting the number of revolutions of the washer.
5. After 10 seconds, record how many revolutions the washer made in your notebook and calculate the orbital period:

1. Let’s say your washer orbited 8 times in ten seconds. We’d calculate the orbital period like this:

1. Repeat with different lengths of twine.
2. Show your results in a chart. Can you find a mathematical relationship between distance and orbital period?

Shorter lengths of twine will have faster speeds and shorter orbital periods. Longer lengths will have high orbital periods and slower speeds. Earth’s orbital period is 365 days/revolution.

Gravity is modeled by the tension in the rope, which keeps the washer from flying away on a tangent. For shorter lengths of twine, you will need to swing the system faster and harder than for longer lengths. This is due to centripetal force in the circle.

This is a decent model for a circular system. In reality, planetary orbits in our solar system are ellipses with two foci, with the sun being at one focus. Kepler’s 3rd Law of Planetary Motion states that the square of the orbital period of a body orbiting around a larger body is proportional to the cube of the semi-major axis of the body’s orbit, which is basically the body’s distance from the larger body. Here’s an easy way to think about it: the more distant a planet is from the sun, the greater its orbital period.

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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.