Kepler’s Laws

! Do the planets move east or west over the course of one night? Over the course of several nights? How do true motion and retrograde motion differ?

! What are geocentric and heliocentric models? What are inferior and superior planets? What are opposition, and inferior and superior conjunction? What is an epicycle?

! Can you sketch Ptolemy’s and Copernicus’ models & how they explain retrograde motion & inferior planets?

! What was Tycho Brahe’s contribution to astronomy? Who was Kepler and what are each of Kepler’s Laws?

! Note the [xtra] beside lectures 1-7 on the schedule. These may be helpful to learn difficult concepts

Learning Objectives

The Planets! In ancient times,

people noted five bright “stars” that moved through the constellations of the Zodiac over time

! These “stars” were called planets, from Greek for “wanderers”

http://antwrp.gsfc.nasa.gov/apod/ap990325.html

MercuryJupiter

Venus

Saturn

Planetary Motion! On a single night, the planets will rise and

appear to move from east to west across our sky (due to the Earth’s rotation)! just like the stars and the Sun

! Over multiple nights, planets move west to east with respect to distant background stars (due to the Earth and the planets all orbiting the Sun counter-clockwise) ! this is their true motion

! Like the Sun, the planets appear to move through the Zodiac, close to the ecliptic

Explaining Planetary Motion

! The Greek philosophers almost all believed that we lived in a geocentric (Earth-centered) system!The heavens move in perfect circles!The celestial domain (the “sphere of

the stars” is pure and unchanging)! The geocentric system held sway for

over 2000 years

Planetary Language! Inferior conjunction

!When a planet passes in front of the Sun

! Superior conjunction!When a planet passes behind the Sun

! Opposition!When a planet is opposite the Sun in

the sky (and is therefore visible in the middle of the night)

Inferior vs. Superior Planets! Venus, Mercury are always close to the Sun

!Venus is always within 46º of the Sun!Mercury is always within 22º of the Sun!Referred to as the inferior planets!Show inferior and superior conjunctions but

no oppositions! Mars, Jupiter, and Saturn are not “tied to

the Sun” in the sky!Referred to as the superior planets!Show oppositions and superior conjunctions

but no inferior conjunctions

Retrograde Motion! A planet's

west-to-east“true” motion with respect to the stars overmultiple nightscan stop and reverse to a slow east-to-west motion

! This is called retrograde motion

Ptolemy (140 AD)! The basic geocentric

model is quite simple! The Earth is in the

center! ! Ptolemy created a

geocentric model with uniform circular motion to explain retrograde motion and inferior planets! The Ptolemaic system

Ptolemy Explains Retrograde Motion

! Planets move on epicycles, which move along a larger circular orbit

! Ptolemy’s system could explain retrograde motion in an Earth-centered system!

Ptolemy Explains the Inferior Planets

! Venus' and Mercury's orbits are locked to the Sun and they just move around their epicycles

! This still doesn’t explain superior conjunctions of Mercury and Venus (which are very hard to see, so Ptolemy may nothave known about them)

! Ptolemy’s model becomes very complex

Nicholas Copernicus (ca. 1540)

! Resurrected the heliocentric model

Copernicus Explains Retrograde Motion

!The inner planets move faster than the outer planets

!Retrograde motion occurs when the Earth overtakes and passes (or is passed by) another planet

Copernicus Explains Inferior Planets

! Inferior planets have orbits closer to the Sun than the Earth’s orbit is

! They always appear close to the Sun as seen from Earth

! This can explain the superior conjunctions of inferior planets

Tycho Brahe (ca. 1580)! Observed a new star (a “nova”)

in 1572 and showed it was farther away than the planets

! At the time the distant heavens (Ptolemy’s “sphere of the stars”) were considered unchangeable

! Brahe compiled very careful observations over decades

! Accurate to 1 arcminute (1/30th the Moon’s size)!And Tycho did all this without a telescope

Johannes Kepler (ca. 1600)! Inherited a set of very

accurate observations of the planets when Brahe (his boss) died

! Used the data to guide the creation of a new heliocentric theory

! The data could not be fit by circular orbits

! Considered an ellipse

Kepler's First Law

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

The heavens do not move in perfect circles!

Kepler's Second LawThe line that connects the planet to the Sun

sweeps out equal areas in equal time

! Planets move faster when closer to the Sun! Fastest at perihelion (closest to the Sun)! Slowest at aphelion (farthest from the Sun)

Kepler's Third LawA planet’s orbital period (P) in years, squared, equals the semimajor axis of its orbit (a) in AUs, cubed

P2 = a3

years AUs astronomical unit

(1 x Earth-Sun distance)

Planetary Orbits! The farther away from the Sun, the

longer it takes for the planet to orbit AND the slower its average orbital speed

Planet P (yr) a (AU) P2 a3

Mercury 0.24 0.39 0.06 0.06Venus 0.61 0.72 0.37 0.37Earth 1.00 1.00 1.00 1.00Mars 1.88 1.52 3.5 3.5

Jupiter 11.86 5.20 141 141Saturn 29.46 9.54 868 868

Kepler's 3rd Law by Newton! Isaac Newton used his Law of

Gravity to derive a general version of Kepler’s 3rd Law

! The extra M (mass) arises as Kepler only considered how the Sun’s gravity affects the planets and ignored the fact that the planets also have a (tiny) gravitational effect on the Sun

(P is in years, a is in AUs, M is in solar masses)! Kepler's Law works because the Sun’s mass is

much larger than any planet’s mass (M1 >> M2)

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