Kepler’s 3 Laws of Planetary Motion

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• Kepler Firsts

How to Draw an Ellipse:

Kepler’s First Law (The Law of Orbits)

• The orbit of a planet is an ellipse where one

focus of the ellipse is the sun.

Kepler’s Second Law (The law of Areas)

• A line that connects a planet to the sun sweeps out equal areas in equal times.

This law arises from conservationof momentum. When the planetis closer to the sun, it moves faster, sweeping through a longer path in a given time.

Kepler’s Third Law (The Law of Periods)(Also known as

Law of Harmonics)The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

T2 = R3

T = Period, R = semi-

major axis

Notice the slope of the

line is the same.

Brahe to Kepler to Newton

Kepler used the data from Brahe’s observations of Marsto come up with his three laws. Newton used Kepler’s

Third Law to derive the Law of Universal Gravitation:

Newton already knew that an object’s gravitationalforce was inversely proportional to the squareof its distance from the Earth’s center.

F(grav) = GMm/r2

From Kepler: T2/R3 = 2.97 x 10-19 s2/m3 for any planet.

Consider the net Centripetal Force = mplanetv2/R

Set them equal to each other GMm/R2 = mv2/R

cancel m and RGM/R = v2

Since v = 2pr/T then substitute in for v2

GM/R = 4p2R2/T2

GM/R = 4p2R2/T2

GM = 4p2R3/T2

4p2/GM =T2/R3

You can test this: M is mass of sun: 1.98 x 1030 kg.

Example problems:

Use these examples to determine if you are using Kepler’s Third Law correctly: – An asteroid orbits the sun at a distance of 2.7 AU. What is its orbital period? • Using a = 2.7 AU, you should get P = 4.44 years.

Note that the distance is in AU. If you have a distance in other units you have to convert them to AU.

– A dwarf planet discovered out beyond the orbit of Pluto is known to have an orbital period of 619.36 years.What is its average distance from the sun? • Using P = 619.36 years, you should get a = 72.66 AU. – Chiron is a dwarf planet that orbits the sun between Saturn and Uranus, and has an average distance from the sun of 14 AU. What is its orbital period? • Using a = 14.0 AU, you should get P = 52.38 years

A solar system called Bizarro World System - 3 has an M class planet that is orbiting its sun at a distance of 13 billion kilometers. What is the orbital period of this planet?

13 E 9 km x _____1AU_____ 86.9 AU149.6 E 6 km

T2 = R3 Square Root of 86.93 = 810 years