Planetary Motion

It’s what really makes the world go around.

What is a year and a day? (write this down!!)

• A year = time it takes for the Earth to travel around the sun once (orbital period)

• A day - how long it takes a planet to spin on its axis

• AU - Astronomical Unit - distance from

Sun to Earth

Satellites

• Satellites are bodies in orbit around a central point.

• The moon is a satellite of Earth and Earth is a satellite of the Sun.

• Most satellites are the result of passing objects that get caught by the pull of a larger body and are not moving fast enough to get away.

Speed : life or death

• Too slow, the satellite will get pulled into the planet and crash into it.

• Too fast, it just shoots on by with a change in its path.

• Just right, it enters into an elliptical orbit around the body.

Drawing an ellipse

• The black line is a piece of string

• The thumb tacks are the “foci” of the ellipse

Johannes Kepler

• 1571-1630• Made many

observations of the motions of Mars

• Supported Galileo• Kepler didn’t know why

his Laws worked—a law summarizes observations, it doesn’t explain the reason!

Kepler’s Laws

How can we describe how the planets move?

Decades of observations + (Math afterwards) = SUCCESS

Kepler’s First Law

• Planets travel in elliptical orbits about the sun

• The sun is at one foci of the ellipse

• Eccentricity (e) is used to describe the shape.

Orbits in the Solar System

• The planets have nearly circular orbits

• Lower e = closer to a circle

• Icarus, an asteroid, has a very elliptical orbit

• Comets have the most elliptical orbits

Halley’s cometDirect Motion: Planet will generally move from East to West across the sky

• Retrograde motion: planet will move from East to West across the sky but occasionally move West to East- caused by differences in speed of orbit

• As we see from the animation, Mars isn’t really moving backwards, but it appears to when viewed from Earth!

An edge on view of the solar system

First Law

• Kepler’s First Law was developed from the data he collected.

• It explained the Retrograde motion observed for planets in the sky.

• It was the first relationship that did not have special conditions for various times of the year.

Kepler’s Second Law• The planets sweep out equal areas in equal times

• T1=T2, A1=A2• The planets move faster on the side of the orbit closer to

the sun• The planets slow as they get further from the sun

2nd Law In Motion• On the right is a

circular orbit• On the left is an

elliptical orbit.• What can you tell

about the speed of the satellite? What about the blue areas?

Kepler’s Third Law

13

2

R

T

• The square of the period of the rotation is proportional to the cube of the semi-major axis of the ellipse.

• For every body orbiting the same foci, the ratios of T2/R3 are equal to each other.

• For all bodies in orbit around the sun,

Semi-major axis

• The arrow represents the semi-major axis.

Kepler’s Third Law

Does Kepler’s 3rd Law Work?

• Astronomical unit = A.U.= average distance between the earth and the sun

– A.U. = 150 million km or 93 million miles

• 1 year = period of Earth’s orbit• T2 = R3 if T is in years and R is in A.U.

Does Kepler’s 3rd Law Work? Prove that T²/R³=1 for each planet below and turn

in for credit. Show at least one set of calculations!

Planet Period T (yr)

Semi-Major Axis R (AU)

T2 R3

Mercury 0.24 0.39 0.06 0.06

Venus 0.62 0.72 0.39 0.37

Earth 1.00 1.00 1.00 1.00

Mars 1.88 1.52 3.53 3.51

Jupiter 11.9 5.20 142 141

Saturn 29.5 9.54 870 868

So What?

• Kepler didn’t know why his laws worked! Remember a law just describes a relationship, it does not tell you why.

• Newton was able to show that he could derive Kepler’s Laws from Newton’s three laws and the theory of universal gravitation.

Newton’s Law of Universal Gravitation

• If one of the masses doubles the force doubles• If the radius doubles the force decreases to 1/4

221

r

mmGF

What does the gravitational force look like as a function of distance?

Universal Gravitation

0

20

40

60

80

100

120

140

160

180

200

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance between objects

Fo

rce

m1,m2=1

m1=1, m2=2

m1=1,m2=10