Kepler
This text is meant to accompany class discussions. It is not everything there is to know about Kepler's Laws. It is meant as a  prep for class. More detailed notes and examples are given in the class notes, presentations, and demonstrations.
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Johannes Kepler developed his three las of planetary motion without the use of computers, calculators, or even a telescope. He made his laws before telescopes were invented. He made his laws from data that wasn't even his. Before Kepler there was another astronomer, named Tyco Brahe, whose collection of precise heavenly data helped Kepler.

Brahe knew that the secret to uncovering the mysteries of the heavens lie in collecting precise data. Brahe designed, built and calibrated astronomical instruments. His instruments and records became the most precise collection of data. Kepler worked under Brahe's tutelage.

Brahe was jealous of Kepler's ability and gave him what appeared to be the hardest data to work with. It was the orbital data for Mars. What made it difficult to analyze was that fact that Brahe was trying to make the data trace our the path of a circle. Mars has the most elliptical orbit of the inner planets. Kepler recognized that this and realized his first law of planetary motion.

The video above is from YouTube. It is Carl Sagan's 6 minute summary of Kepler's 3 Laws. The web address is http://youtu.be/XFqM0lreJYw

 

1st Law

All orbits are ellipses with the gravitational source at one or more foci.
A circle is a special case of an ellipse. This is why circular orbits meet the rule. See the animation below.

This law is also known as the "Law of Orbits."

 

2nd Law

Equal areas are swept out during equal time intervals.

The video above shows an animation of the 2nd law from YouTube at http://www.youtube.com/watch?v=_3OOK8a4l8Y. Each shaded region occurs at an equal segment of time. The time interval in this animation is a month. So the shaded region is created every month. The area of each region is the same.

The implication of this is that a orbiting planet moves faster when it is closer to the star and slower when it s farther away from a star.

While this law created from the observations of the planets, it applies to everything that orbits in the universe.

 

This law is also knwon at the "Law of Areas."

 

Kepler's 3rd Law


Objects that orbit other bodies are called satellites. There are natural satellites like the Moon or a planet and there are artificial satellites that mankind has launched into orbit. The "body" is the object being orbited. Early on in history, all distances were measured in astronomical units (abbreviated as "au.") An astronomical unit is the average distance between the Earth and the Sun. 1 au equals 149 597 870 691 ± 30 meters.

Kepler's constant is a mathematical law relating the orbits of all the satellites that orbit the same body.

"r," the radius of the orbit, is measured from the center of the satellite to the center of the body. "T," the period of motion, is the time for the satellite to orbit the body once. One of the nice aspects of this law is the fact that the units of the period and radius do not matter as long as they match on both sides of the equation. Comparing Kepler's constant between systems does not indicate anything about the systems. This law (as applied here) assumes the orbits are circular. (See below for more about what the third law really says and how is applies to ellipses.)

  • Example
  • Hint
  • Solution
Callisto is a moon of Jupiter. It takes 16.6890184 Earth days to orbit Jupiter. Ganymede is another moon of Jupiter. It takes Ganymede 7.15455296 Earth days to orbit Jupiter. Ganymede is measured to be 1,070,000,000 m from Jupiter's center. How far away is Callisto from the center of Jupiter.

Callisto is a moon of Jupiter. It takes 16.6890184 Earth days to orbit Jupiter. Ganymede is another moon of Jupiter. It takes Ganymede 7.15455296 Earth days to orbit Jupiter. Ganymede is measured to be 1,070,000,000 m from Jupiter's center. How far away is Callisto from the center of Jupiter.


Hint

This is a great example of when to use Kepler's third law to solve a problem. When you are given to satelites orbiting the same body and you're given 3 out of four of distances from the center of the body to the satelites and the period you can use Kepler's third law. These bodies all have the same Kepler constant because they revolve around the same body.

Kepler's third law

Since both of these setellites are orbiting the same body, Jupiter, and therefore have the same Kepler's constant. Therefore

Ratio

This is the typical set up for solving problems using Kepler's third law. Because everything is set up as a ratio, you do not need to convert units to any known standard. But you do need to make sure they all have matching units.

 

Callisto is a moon of Jupiter. It takes 16.6890184 Earth days to orbit Jupiter. Ganymede is another moon of Jupiter. It takes Ganymede 7.15455296 Earth days to orbit Jupiter. Ganymede is measured to be 1,070,000,000 m from Jupiter'e center. How far away is Callisto from the center of Jupiter.


Hint (scroll down for the solution)

This is a great example of when to use Kepler's third law to solve a problem. When you are given to satelites orbiting the same body and you're given 3 out of four of distances from the center of the body to the satelites and the period you can use Kepler's third law. These bodies all have the same Kepler constant because they revolve around the same body.

Kepler's third law

Since both of these setellites are orbiting the same body, Jupiter, and therefore have the same Kepler's constant. Therefore

Ratio

This is the typical set up for solving problems using Kepler's third law. Because everything is set up as a ratio, you do not need to convert units to any known standard. But you do need to make sure they all have matching units.


Solution

Unless otherwise stated all distances are between the center's of the satellite and the body it is orbiting for planetary motion problems.

Use Kepler's 3rd law of planetary motion to set up a ratio problem comparing the period and radii of the two moons.

 

 

 

When solving problems using Kepler's 3rd law of planetary motion keep in mind some givens you may not realize you know. When comparing satellites orbiting our sun, they can be compared to the Earth whose distance is 1 au from the sun and whose period of revolution around the Sun is 1 year. Satellites orbiting the Earth can be compared to the period and distance for the Moon. This can be found in various data tables.

In addition to being called the "Harmonic Law," this law is also refered to as the "Law of Periods."

More about Kepler's 3rd Law


Earlier it was stated, "...that the 3rd law, (as applied here,) was only for circular orbits." However, Let's extend this a little more into the real world. Ellipses are also governed by the third law. (A circle is a special case of an ellipse. For a circle, the semi-major axis, semi-minor axis, and the radius are all equal.) Now the radius in the third law is replaced with the semi-major axis of the ellipse.

 

     

by Tony Wayne ...(If you are a teacher, please feel free to use these resources in your teaching.)