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Planetary Motion

Dr. Bill Pezzaglia

Lecture 8, 9, 10 [Feb 9, 14, 16, 2006]

VI. Planetary Motion (& Gravity)

A. Cycles of Planets

B. Greek Geocentric Model

C. Heliocentric Model

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A. Cycles of Planets

1) The Wanderers

2) Superior Planets

3) Inferior Planets

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The School at Athens

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1a. The Wanderers 5

• The Babylonians discovered that the 5 brightest stars wandered the sky.

• Mercury, Venus, Mars, Jupiter & Saturn• They followed the ecliptic (like sun and

moon), but had erratic paths.

1b. Retrograde motionBabylonians had noted “wandering” motion of planets. Here in 1997 Mars moves “direct” (normal eastward motion) most of the time, but then moves “retrograde”(westward) for awhile, then returns to direct motion.

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1c. Synodic Period

Babylonians note retrograde pattern repeats every “synodic period”, but in a different part of sky.

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378Saturn399Jupiter780Mars

Mayan Calendar584Venus116Mercury

NoteSynodic (days)Planet

2. Superior Planets• Superior Planets: (Mars, Jupiter, Saturn)• They go retrograde when they are opposite the

sun

• Different than “Inferior Planets” which are never seen far from the sun.

• Ancients have no explanation for this behavior

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2a. Superior Planet Configurations

•Further from sun than earth•Opposition is 180 degrees elongation, best time to view (closest, transits at midnight)

9 2b. Synodic Period of MarsHere from July 2005 through February 2006, Mars moves across the constellations Pisces, Aries, and Taurus. Mars's motion is direct (from west to east, or from right to left in this animation) most of the time but is retrograde (from east to west, or from left to right in this animation) during October and November 2005.

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2c. Synodic Period of Mars

• Every 780 days (synodic period) the retrograde motion repeats, but in a different part of the sky.

• Two periods later, in late 2009, Mars moves across the constellations Gemini and Cancer.

11 3a. Inferior Planets• Inferior Planets:

(Mercury, Venus) are never seen far from the sun. They go retrograde at “inferior”conjunction with the sun.

• In “modern”heliocentric model, we can see why the inferior planets never can be opposite the sun.

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3b. Venus Diaries 13

Omen texts, first Babylonian Dynasty (1900-1660 BC), Venus is both the morning and evening star. Tablets of Ammizaduga1650 BC were lost, but copied at Nineveh 650 BC, show 21 years of Venus data, including an 8 year cycle.

Note 5 synodic periods is exactly 8 years, matches 13 orbits of Venus around the sun. Hence Venus will be in the same constellations, repeating the same patterns every 8 years!

3c. Venus Cycles 14

Invisible for 66 days around superior conjunction,

evening star for 254 days

invisible for 9 days (on average) at inferior conjunction

morning star for 254 days

cycle repeats in 584 days

Mayan Indians discovered similar cycle (much later).

3d. Mayan Calendar 15

They identified Venus with the godQuetzalcoatl.

Chichen Itza, Yucatan (1000 A.D.)

B. Geocentric Models

1) Early Greek Models

2) Epicycles

3) The Geocentric Universe

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1. Early Greek Models

a) Plato’s Academy

Plato (427-348 BC) leaves the problem of planetary motion for his students to solve. He insists it must be done with circles

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“Let none but geometers enter here”

1b. Eudoxus (408-355 BC)

introduces “Homocentric Spheres”, a complex system of axes and balls. It generates a retrograde path.

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The “hippopede” path

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extends model, has over 55 spheres to explain retrograde motion of 5 planets

1c. Aristotle (384-322 BC) 19 1d. Herakleides of Pontus (388-310 BC) 20

•First to propose that the earth rotates•Resistance to this idea: wouldn’t falling objects fall sideways due to spin?•Suggests the planets follow paths with “loops”

•NEED A PICTURE HERE

2. Epicycles(a) Appolonius of Perga (262-190 BC) proposes

epicycle concept to explain retrograde motion (also introduces idea of eccentric)

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2c. Hipparchus of Rhodes (190-120BC)extends epicycle ideas for sun (to explain seasons not being same length) and moon’s change in orbital speed. Had include idea that earth is not at center but “eccentric”

23 3a. Claudius Ptolemy

• Claudius Ptolemaeu (87-150 A.D.).

• Refines the epicycle theory.• “Geocentric Model” the earth

is at the center of the universe

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3b. Ptolemy’s Geocentric Universe 25

One of the “flaws” is trying to explain why Mercury and Venus are never seen very far from the sun. The dashed line represents some unexplained connection that keeps them all in line.

3c. Ptolemy’s Geocentric Universe 26

The Ptolemaic system is used for over 1500 years!

C. Heliocentric Models

1) The Revolution

2) Tycho and Kepler

3) Newton and Halley

27 1. The Revolution

a) Aristarchus of Samos (310-230 BC) had calculated that the sun was much bigger than the earth. He proposed that therefore it was more important and should be the center of the solar system. However, parallax of stars was not observed, and it seemed silly to say the earth moved.

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1b. Heliocentric Models 28

1b. NicolausCopernicus

(1473-1543 AD)Copernicus found the old model of Ptolemy gave incorrect positionsof the planets.

He developed the Sun-centered (heliocentric) view of the Universe, which improved the predictionsof planetary positions.

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Fig 1-15, p.34

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1b. Copernican System

• Instead of having 5 deferents with 5 epicycles, you only need 5 circles for the planets.

• The only thing that orbits the earth is the moon.

33 One of the most important books ever …Nicolaus Copernicus “On the Revolution of Heavenly Spheres” (1543)

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35The 1000 Zlotych bill features Copernicus. Due to inflation, it was worth about 10 cents USD when I was last in Poland

1c. Retrograde MotionThe Earth travels around the Sun in a smaller orbit than Mars, and moves more

rapidly than Mars. Consequently, as the Earth overtakes and passes this slower-moving planet, Mars appears for a few months to fall behind and move backward with respect to the background stars.

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1d. Synodic vs SiderealThe synodic period of a planet is the time that elapses between two successive configurations, as seen from Earth - from one opposition to the next, for instance, or from one conjunction tothe next. Because the Earth moves while the planet is moving, a planet's synodic period is not equal to its sidereal period (the time to complete one orbit around the Sun).

37 2a. GalileoGalilei

(1564-1642)A contemporary

of KeplerGalileo was one of thevery first scientists todo experiments tounderstand Nature

He was the firstastronomer to use a telescope (in 1610)to study the sky.

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2a.1 Inventor of Telescope?1608 Hans Lippersheyinvented the telescope1609 Galileo made his own, and let everyone think that he invented it (won an award!)

39 2a.2 Mountains of the MoonGalileo’s drawings of the Moon with his telescope, 1610

He measured the heights of mountains from the shadows cast and found them to be about same size as mountains on the earth!

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2a.3 Milky Way made of stars 41

The “Milky Way” appears as a faint milky white across the sky.

With his telescope, Galileo could magnify it and show it was made of stars!

2a.4 Jupiter has MoonsGalileo saw four little “stars”moving with Jupiter across the sky, and changing their positions every night.

He quickly and correctly concluded that they are in orbit around Jupiter.

This confirmed the Copernican view that the Earth is NOT the center of all motion in the Universe.

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Saw spots on the sun that moved, interpreted that the sun rotates in 28 days

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Galileo saw that Venus was bigger at crescent and smaller

when full.

2a.6 Venus Has Phases 44

2a.6 Venus Phases Prove Heliocentric 45

Galileo saw that the phases were inconsistent with the geocentric model, proved Venus goes around the sun!

2a.6 Venus Phases Prove Heliocentric 46

Galileo saw that the phases were inconsistent with the geocentric model, proved Venus goes around the sun!

2a.6 Venus Phases Prove Heliocentric 47 2a.7 Laws of Inertia 48

Aristotle: objects remain in motion only because a force acted constantly on them (e.g. the air pushes the cannon ball)

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2a.7 Laws of Inertia 49

Aristotle: objects remain in motion only because a force acted constantly on them (e.g. the air pushes the cannon ball)

Galileo: objects will stay in motion UNLESS acted upon by a force.

2a.8 Galileo’s Experiment at Pisa• 1590 Galileo’s Principle:

All bodies fall at the same rate, regardless of mass

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"Men are like wine flasks," he once said to a group of students. "...look at....bottles with the handsome labels. When you taste them, they are full of air or perfume or rouge. These are bottles fit only to pee into!"

-Galileo expressing what he thought of the other faculty members of the University of Pisa.

Needless to say, his teaching contract was not renewed!

2a.8 Law of Falling 51

Aristotle: Heavier balls will fall

faster

Galileo: They fall at the same

rate!

2a.9 Law of Relativity (1632) 52

Aristotle: if the earth was moving (around the sun) we’d feel it. Objects would not fall straight down.

Galileo: Motion is relative. A ball dropped from the crow’s nest will hit at the base of the ship’s mast, even if the ship is moving.

2b. Tycho

•Tycho Brahe measuring star positions (without a telescope)•Measurements of position of Mars showed deviations from Copernican model!•He built a big observatory with gigantic protractors (no telescopes yet!)

53 2b.2 Tycho Brahe’s Uraniborg Observatory 54

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2b.3 Tycho Brahe (1546-1601)

•He suggested a weird hybrid model where planets go around sun, but sun goes around earth

55 2c. Johannes Kepler (1571-1630)

•Tycho at first invited Keplerto help in analysis of his data, but then jealously wouldn’t let him have the information.

•On his deathbed he gave Kepler the data.

•Kepler used it (particular data on Mars), to develop three laws of planetary motion.

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2c.1 Kepler’s 1st Law: Orbits are Ellipses

1605: Kepler realized that the motion of Mars could not be explained with a circular orbit, or the multiple circles proposed by Ptolemy.

He accepted Copernicus’ view that Mars was in orbit around the Sun, rather than around the Earth.

He experimented (mathematically) with orbits of various shapes, and found that Mars’ orbit best fits an ellipse.

57 2c.1 Kepler’s 1st Law (1605)

• Law No. 1. Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus.

– This is contrary to the earlier belief that the orbits were perfect circles or combinations of circles.

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Fig 2-3, p.45

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• Ellipses, circles (parabolas and hyperbolas) are “conic sections”, studied first by the greeks.

• But it would NEVER occur to the greeks that an orbit is an ellipse. (why?)

Fig 2-4, p.45

Focus Focus

Drawing an ellipse

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The Ellipse

Do you remember any of this from high school geometry?

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Focus Focus

Highly eccentric

Not very eccentric

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Fig 2-10, p.53

Planet orbits tend to havelow eccentricity (nearlycircular).

Comet orbits tend tobe highly eccentric.

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2c.2 Kepler’s 2nd Law (1609)

Kepler also noticed that whenMars is closest to the Sun in its elliptical orbit, it moves faster than when it is fartheraway.

This led him to formulate hisSecond Law of Planetary Motion.

65According to his second law, a planet moves fastest when closest to the Sun (at perihelion) and slowest when farthest from the Sun (at aphelion). As the planet moves, an imaginary line joining the planet and the Sun sweeps out equal amounts of area (shown as colored wedges in the animation) in equal intervals of time.

562c.2 Kepler’s 2nd Law (Equal areas in Equal Times) 66

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2c.2 Kepler’s 2nd Law 67 2c.3 Kepler’s 3rd Law: “Harmonic Law” 68

Planets closer to the sun move faster.

This is consistent with his 2nd law, that showed a planet will move faster at perihelion.

He searched for a relationship between orbital period and distance to the sun.

2c.3 Kepler’s 3rd Law (1618)

• The square of the orbital period (P) is directly proportional to the cube of the semimajor axis of the orbit (a).

P2 = a3

This law explains the proportions of the sizes of the orbits of the planets and the time that it takes them to make one complete circuit around the Sun.

[Note: in physics, the symbol “a” is also used to represent “acceleration”. Confused?]

Why is it called the “harmonic law”? Kepler thought the spacing between planets was related to musical intervals.

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An example of Kepler’s third law: The orbit of Mars(Recall: P2 = a3)

Mars’ orbit period (P) is 1.88 years. P2 = 3.53

Kepler’s law says that P2 = a3, so 3.53 = a3. So then a = (3.53)1/3 (the cube root of 3.53), or 1.52.

Thus, the semimajor axis (average distance of Mars from the Sun) is 1.52 Astronomical Units.

But how big is an Astronomical Unit?

Kepler didn’t know.

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c. The distances of the planets from the Sun

• In the Copernican world view, the planets are in orbit around the Sun.

• Astronomers knew the relative distances of the planets, but not the absolute distances.

• Known: Jupiter is 5 times farther from the Sun than the Earth is. It takes Jupiter 12 times longer to go around the Sun than it does for the Earth.

• Not known: How many kilometers (or miles) are the Earth and Jupiter from the Sun?

• Fundamental Question: What is the absolute scale of the Solar System?

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“How large is the Astronomical Unit?”

Astronomical Unit (AU) The average distance from the Earth to the Sun

150,000,000 kilometers, or 93,000,000 miles

But how was this measured ?

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• The size of the Astronomical Unit was derived over many years, in part by observations made on voyages by Captain James Cook (1768-70)

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3. Isaac Newton (1643-1717)Why do the planets move in the ways described by Kepler?

75a). Newton’s Three Laws formulated

around 1666 (not published for 20 years!)• 1st Law: (adopted from Galileo’s laws of inertia),

– bodies at rest tend to stay at rest, bodies in motion tend to stay in constant motion,

– unless compelled to change by an outside force.– “Conservation of momentum” (momentum=mass x velocity)

• 2nd Law: Force = mass x acceleration– mass “m” is measure of inertia (kilograms)– acceleration “a” is the change in motion (meters/second2)– “Force” is measured in units of “Newtons”

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-Newton’s Laws Continued-

• 3rd Law: “To every action there is an equal and opposite reaction”– This means, as much as the earth is pulling on you– You are pulling back on the earth the same amount– Without this law, energy and momentum would

not be conserved in the universe.

• 4th Law: Law of Gravity came later (1680)

77 Christiaan Huygens (1629-1695)

• Around 1654 Huygens devised a new and better way of grinding and polishing telescope lenses.

• In 1655, he discovered first moon of Saturn (Titan). The following year he discovered the true shape of the rings of Saturn, explaining phases and changes in shape.

• In 1656 he patented the first pendulum clock, which greatly increased the accuracy of time measurement.

• In 1673 he published he law of centrifugal force for uniform circular motion. As a result of this Huygens, Hooke, Halley and Wren formulated the inverse-square law of gravitational attraction (before Newton).

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•Acceleration=(velocity)2/radius = (4π)2 (Radius)/(Period)2

•Recall Kepler 3rd Law: (period)2 proportional to (radius)3

•Implies acceleration of gravity is inversely proportional to squared distance: acceleration =~ 1/R2

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The History of Newton’s 4th Law of Gravity• In 1679 Hooke writes Newton, proposing inverse square law.

• In 1684 Wren, Hooke and Halley discussed, at the Royal Society, whether the elliptical shape of planetary orbits was a consequence of an inverse square law of force depending on the distance fromthe Sun.

• Later in the same year in August, Halley visited Newton in Cambridge and asked him what orbit a body would follow under an inverse square law of force

• Sr Isaac replied immediately that it would be an Ellipsis, the Doctor struck with joy and amasement asked him how he knew it, why, said he I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, Sr Isaac looked among his papers but could not find it, but he promised him to renew it, and then to send it him.

• Actually, he had apparently derived it in 1680, after correspondence from Hooke. Later Hooke claims Newton stole the idea and did not give him credit!

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•Halley is the one that got Newton to write the Principia (1687). In fact paid for the publication out of his own pocket.

•1695 Halley made a careful study of the orbits of comets, proposing them to be elliptical (Newton thought they were parabolic). He calculated that the comet of 1682 was the same as the comet of 1607 and 1531,i.e. appears every 76 years.

•1705 redicts that it will appear in December 1758. Itappears on December 25, some 15 years after his death.

Edmond Halley(1656-1742)

• 1676 Sails to St. Helena, maps the southern sky• Proposes using transits of Mercury & Venus

(the timing of planet passing across the face of sun) to determine the astronomical unit (done much later by James Cook 1768)

• Shows that Kepler's third law implied the inverse square law of attraction and presented the results at a meeting of the Royal Society on 24 January 1684

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Newton’s Law of Gravity 81

•In the Principia Newton also deduced Kepler's third law.

•Huygens criticized the law. How can one believe that two distant masses attract one another when there is nothing between them? Nothing in Newton's theory explains how one mass can possible even know the other mass is there. ("action at a distance")

A body exerts a force on other bodies around it, eventhough they are not touching. That force is related to mass ofeach body (M1 and M2) and the distance separating them (R).

Force due to gravity = G M1M2 / R2

G is the “gravitational constant”, measured 100 years later by Cavendish

The Newton-Kepler Law• In the Principia Newton also deduced Kepler's

third law, but in an important new form

• Mass of central body: M = a3/P2

– Orbital Radius “a” (in astronomical units)– Period “P” (in years)– Mass “M” in units of “solar masses”

• To measure mass of– Earth, use moon’s orbit– Jupiter, use Galilean moons– Sun, use orbits of planets– Galaxy, use orbits of stars around galaxy

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Fig 2-11, p.54

Galileo proposed that throwing a ball at different speeds causes it to travel farther before it falls to Earth.

Throw it fast enough, and as it falls the earth’s curve falls underneath it, and it falls forever (“free fall”)

The critical speed is called Orbital Velocity.

For Earth, orbital velocity is 17,500 miles/hr, or 8 km/sec

Orbital velocity

83Escape velocity

Throw the ball hard enoughand it will escape thegravity pull of the Earth

How fast do you have tothrow it?

It depends on which planetyou live on …

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Escape Velocity for the planets

rGmV /2=

The velocity of escape, V, from a planet is given by

G is the universal gravitational constantm is the mass of the planetr is the radius of the planet

The escape velocity of Earth is 11 km/sec, or About 25,000 miles per hour.

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2.40.270.012Moon4.30.380.05Mercury5.20.520.11Mars10.40.960.8Venus22.43.715Uranus25.43.417Neptune35.69.595Saturn60.011.2318Jupiter620 km/sec109330,000Sun

Escape Velocity

Radius (Earth = 1)

Mass (Earth = 1)

Object

Escape Velocities for the planets, the Sun, and the Moon 86

(e1). Special Relativity 87

1905 Einstein (26 years old) publishes theory of special relativity

•Speed of light is the same for all observers

•Motion is relative (Galileo)• there is no experiment one can do to determine absolute motion relative to “space”.

Laws of physics must hold in all reference frames which differ only by a constant velocity

e2. Galileo’s Experiment at Pisa• 1590 Galileo’s Principle: all

bodies fall at the same rate, regardless of mass

• 1907 Strong EEP(Einstein Equivalence Principle)same result, but argued from a different way.

• He proposed that falling bodies in gravity are equivalent to being in an accelerated frame (e.g. in an accelerating elevator)

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e3. The Equivalence PrincipleReference at rest with Gravity is indistinguishable to a reference frame which is accelerating upward in gravity free environment.

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The apple accelerating downward due to gravity looks the same as an apple at rest in space, with the floor accelerating upward towards it.

References90

Good Java Demo: http://www.jimloy.com/cindy/galilean.htm