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Lesson Objectives • Describe Kepler's three laws of planetary motion. • Solve problems using Kepler's laws. • Explain the effects of Earth, the moon, and the Sun on each other.

Earth-Moon-Sun System Section 6-2

Kepler’s First Law

Johannes Kepler

• The orbits of planets are ellipses with the Sun at one focus

• 1571-1630 • Built on Nicolaus Copernicus’s heliocentric model of the solar system • Analyzed twenty years of data collected by Tycho Brahe • Developed three laws of planetary motion

Ellipses

Elliptical Orbit Vocabulary

Co-vertex Minor Axis Vertex

Major Axis Focus

Center

Focus

Vertex

• Perihelion: The point on an object’s orbit closest to the star • Aphelion: The point on an object’s orbit furthest from the star • Semi-major Axis (): Half of the longer axis – Half the distance from perihelion to aphelion – Average distance an object is from the star it orbits

Co-vertex

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Kepler’s Second Law

Kepler’s Second Law

• The speed of a planet varies, such that it sweeps out equal areas in equal time frames.

Kepler’s Second Law

Kepler’s Third Law

• Distance from sun changes, since elliptical • Gravitational force changes, since  = • Tangential speed changes, since  =

  

 

• The square of the orbital period, , is directly proportional to the cube of the average distance from the Sun, . ∝  • Comparing two objects orbiting the same central body:

   =  

Newton’s Version of Kepler’s 3rd Law

You-Try #1 • Use the data provided to determine the time it takes Mars to orbit the Sun. • Earth’s orbital period = 1.00 years • Earth’s average distance from the Sun = 1.00 AU • Mars’s average distance from the Sun = 1.52 AU

= • • • •

4  

= orbital period  = universal gravitational constant  = mass of the central body = semi-major axis of the orbit (the average distance between the orbiting object and the central body)

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Rearranging =

   

You-Try #2

4  

   = 4

• Saturn’s moon Mimas has an orbital period of 82,800 s at a distance of 1.87 × 10" m from Saturn. Determine Saturn’s mass.

Simplified Kepler’s 3rd Law

You-Try #3

 =



∙ $%& = • = orbital period, measured in Earth years • $%& = mass of the star, measured in solar masses, '⊙

• Determine the orbital period of Venus, measured in Earth years. The semi-major axis of Venus’s orbit is 1.08 × 10-- m = 0.723 AU.

– 1 '⊙ = 1.989 × 10+ kg = mass of the sun

• = semi-major axis of the orbit, measured in astronomical units, AU. • An astronomical unit (AU) is the average distance from the Earth to the Sun, 1.496 × 10-- m.

You-Try #4 • Saturn orbits the Sun at an average distance of 1.43 × 10- m. Determine Saturn’s orbital period, measured in Earth years.

You-Try #5 • An exoplanet is observed orbiting a 5.7 solar mass star at an average distance of 3.2 AU. What is the orbital period of this exoplanet?

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You-Try #6 • What is the orbital period of the moon around Earth? Earth’s mass is 5.97 × 10  kg. The semi-major axis of the moon’s orbit is 3.83 × 10" m. Convert your answer to days.

Earth and Moon • Moon’s orbit around Earth is nearly circular. • Has equal rotational and orbital periods due to tidal locking. • This is why the same side of the moon is always facing Earth.

You-Try #7 • What is the moon’s rotational period? The radius of the moon is 1.76 × 104 m. The rotational tangential speed of the moon is 4.63 m/s. Convert your answer to days.

Tides • The Moon pulls the closer side of the Earth more strongly than the far side of the Earth. • This stretches out the Earth’s oceans. • As Earth rotates, the stretching changes, leading to ocean tides. • The sun has a similar, but smaller, effect.

Unit 6 Assignments • Read textbook chapter 14 (pages 199-209) – Page 209 #1-8, 11-17

• 6-1 Practice Problems • 6-2 Practice Problems

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