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Astro110-01 Lecture 8The Copernican Revolution

(Cont’d)

or the revolutionaries:Nicolas Copernicus (1473-1543)

Tycho Brahe (1546-1601)Johannes Kepler (1571-1630)

Galileo Galilei (1564-1642)Isaac Newton (1642-1727)

who toppled Aristotle’s cosmos

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Johannes Kepler(1571–1630)

• In the interplay between quantitativeobservation and theoretical construction thatcharacterizes the development of modernscience, Brahe was the master of the first butwas deficient in the second.

• The next great development in the history ofastronomy was the theoretical intuition ofJohannes Kepler (1571-1630), a German whowent to Prague to become Brahe's assistant.

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Kepler and the Elliptical Orbits• Unlike Brahe, Kepler believed firmly in the

Copernican system.

• Kepler realized that the orbits of the planets werenot the circles but were instead the "flattenedcircles" called ellipses

The difficulties with the Martian orbit deriveprecisely from the fact that the orbit of Mars wasthe most elliptical of the planets for which Brahehad extensive data.

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An ellipse looks like an elongated circle.

What is an ellipse?

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Eccentricity of an Ellipse

Eccentricity and Semimajor Axis of an Ellipse

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Kepler’s three laws of planetarymotions

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Kepler’s First Law:The orbit of each planet around the Sun is anellipse with the Sun at one focus.

[Greek:away fromthe Sun]

[Greek: nearthe Sun]

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Kepler’s Second Law:As a planet moves around its orbit, it sweepsout equal areas in equal times.

A planet travels faster when it is nearer to theSun and slower when it is farther from the Sun.

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Kepler's 2nd Law

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Kepler’s Third Law• The ratio of the squares of the revolutionary periods for two

planets is equal to the ratio of the cubes of their semimajor axes:

• Choosing subscript 1 for the Earth, the relation can be rewrittenas:

p2 = a3

with p = orbital period in years and a = average distance from Sun in AU

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Kepler's Third Law implies that the period fora planet to orbit the Sun increases rapidly withthe radius of its orbit.

More distant planets orbit the Sun moreslowly than the ones that are closer

- Mercury, the innermost planet, takes only88 days to orbit the Sun

- the outermost planet (Pluto) requires 248years to do the same.

Kepler’s Third Law

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Planets’ period

http://www.ac.wwu.edu/~stephan/Astronomy/planets.html

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Kepler’s Third Law (Cont’d)

• The only thing that affects the orbitalperiod p of the planets is the semimajoraxis a

• The mass, and orbital eccentricity, donot matter

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Kepler’s Third Law

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Graphical version of Kepler’s Third Law

This graph shows that Kepler’s 3rd lawhold true. The graph shows theplanets that were known duringKepler’s time

This graph shows how theorbital speeds of the planetsdepend on their distances fromthe Sun: More distant planetsorbit the Sun more slowly.

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Clicker Question

A. 4 yearsB. 8 yearsC. 16 yearsD. 64 years

(Hint: Remember that p2 = a3.)

An asteroid orbits the Sun at an average distancea = 4 AU. How long does it take to orbit the Sun?

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An asteroid orbits the Sun at an average distancea = 4 AU. How long does it take to orbit the Sun?

A. 4 yearsB. 8 yearsC. 16 yearsD. 64 years

We need to find p so that p2 = a3. Since a = 4, a3 = 43 = 64.Therefore p = 8, p2 = 82 = 64.

Clicker Question

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Clicker question: Planetary orbits

When we say that a planet has a highlyeccentric orbit, we mean that:

1. it is spiraling in toward the Sun.2. its orbit is an ellipse with the Sun at

one focus.3. in some parts of its orbit it is much

closer to the Sun than in other parts.

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Clicker question: Planetary orbits

When we say that a planet has a highlyeccentric orbit, we mean that:

1. it is spiraling in toward the Sun.2. its orbit is an ellipse with the Sun at

one focus.3. in some parts of its orbit it is much

closer to the Sun than in otherparts.

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Clicker question: CometsSuppose a comet orbits the Sun on a highly eccentric

orbit with an average (semimajor axis) distance of 1AU. How long does it take to complete each orbit,and how do we know?

1. It depends on the eccentricity of the orbit, asdescribed by Kepler's second law.

2. 1 year, which we know from Kepler's third law.3. Each orbit should take about 2 years, because the

eccentricity is so large.4. It depends on the eccentricity of the orbit, as

described by Kepler's first law.

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Clicker question: CometsSuppose a comet orbits the Sun on a highly eccentric

orbit with an average (semimajor axis) distance of 1AU. How long does it take to complete each orbit,and how do we know?

1. It depends on the eccentricity of the orbit, asdescribed by Kepler's second law.

2. 1 year, which we know from Kepler's third law.3. Each orbit should take about 2 years, because the

eccentricity is so large.4. It depends on the eccentricity of the orbit, as

described by Kepler's first law.

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Problem 1

• The recently discovered object Eris,which is slightly larger than Pluto, orbitsthe Sun every 560 years. What is itsaverage distance (or semimajor axis)from the Sun?

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Problem 1: solution

Use Kepler’s 3rd law to find the period:p2 = a3

Solve for a:a = p2/3

Take p = 560 yra = 67.9 AU

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Problem 2

• Halley’s comet orbits the Sun every 76years and has an orbital eccentricity of0.97– Find its average distance to the Sun (i.e. its

semimajor axis)

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Problem 2: solution

Use Kepler’s 3rd law to find the period:p2 = a3

Solve for a:a = p2/3

Take p = 76 yra = 17.9 AU

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Problem 3

Halley’s orbit is very eccentric (stretched-out ellipse), so that at perihelion it isonly about 90 million km from the Sun,compared to more than 5 billion km ataphelion.– Does Halley’s comet spend most of its time

near its perihelion, aphelion, or halfwaybetween?

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Problem 3: Solution

Halley’s comet spends most of its time far fromthe Sun near aphelion; since Kepler’s secondlaw says that bodies move faster when theyare closer to the Sun than when they arefarther away.

Halley’s comet moves most slowly ataphelion.

Since it is moving most slowly there, itspends more time in that part of the orbit

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Galileo Galilei (1564–1642)

The main objections of the Aristotle viewto a Sun-centered Universe were:

• Earth could not be moving becauseobjects in air (birds, clouds, ..) wouldbe left behind as Earth moved alongits way

• Noncircular orbits are not “perfect”as heavens should be

• If Earth were really orbiting the Sun,we would detect stellar parallax

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Overcoming the first objection(nature of motion):

Galileo’s experiments with rolling balls and droppingobjects from a height showed that objects in airwould stay with a moving Earth.

Aristotle thought that all objects naturally come torest.

• Galileo showed that objects will stay in motionunless a force acts to slow them down (Newton’sfirst law of motion).

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Overcoming the second objection(heavenly perfection)

Tycho’s observations of a comet anda supernova already challengedthis idea.

• Using his telescope, Galileo saw:

— Sunspots on Sun(“imperfections”)

— Mountains and valleys on theMoon (proving it is not a perfectsphere)

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• Tycho thought he had measured stellardistances, so lack of parallax seemed to rule outan orbiting Earth.

• Galileo used his telescope to see that the MilkyWay is made of countless individual stars:

showed that stars must be much fartherthan Tycho thought.

If stars were much farther away, then lack ofdetectable parallax was no longer so troubling.

Overcoming the third objection (parallax)

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The final nails in the coffin of thegeocentric model

• Two of Galileo’s earliest discoveriescontributed to the demise of thegeocentric model

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Galileo thus proved thatnot all objects orbit Earth.

1. Galileo’s discovery of four moons orbiting Jupiter

Page from Galileo’s notebookwritten in 1610. His sketchesshow four “stars” near Jupiter(the circle) but in differentpositions at different times (andsometimes hidden from view).

Galileo soon realized that the“stars” were actually moons.

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2. Galileo’s observations of phases of Venus proved that Venus orbits the Sun and not Earth.

In the Ptolemaic model, Venusorbits Earth, moving around asmaller circle on its larger orbitalcircle; the center of the smaller circlelies on the Earth-Sun line. If thisview were correct, Venus’ phaseswould range only from new tocrescent

In reality, Venus orbits the Sun, sofrom Earth we can see it in manydifferent phases. This is just whatGalileo observed, allowing him toprove that Venus orbits the Sun.

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GalileoGalileo observed all of the following. Which

observation offered direct proof of a planetorbiting the Sun?

1. Phases of Venus2. The Milky Way is composed of many

individual stars3. Four moons of Jupiter4. Patterns of shadow and sunlight near the

dividing line between the light and darkportions of the Moon's face

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GalileoGalileo observed all of the following. Which

observation offered direct proof of a planetorbiting the Sun?

1. Phases of Venus2. The Milky Way is composed of many

individual stars3. Four moons of Jupiter4. Patterns of shadow and sunlight near the

dividing line between the light and darkportions of the Moon's face

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In 1633 the CatholicChurch ordered Galileo torecant his claim that Earthorbits the Sun.

His book on the subjectwas removed from theChurch’s index of bannedbooks in 1824.

Galileo was formallyvindicated by the Churchin 1992.

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What have we learned?• How did Copernicus, Tycho, and Kepler challenge

the Earth-centered idea?— Copernicus created a Sun-centered model;— Tycho provided the data needed to improve this model;— Kepler found a model that fit Tycho’s data.

• What are Kepler’s three laws of planetary motion?1. The orbit of each planet is an ellipse with the Sun at one

focus.2. As a planet moves around its orbit it sweeps our equal

areas in equal times.3. More distant planets orbit the Sun at slower average

speeds: p2 = a3.

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What have we learned?

• What was Galileo’s role in solidifying theCopernican revolution?— His experiments and observations overcame the

remaining objections to the Sun-centered solarsystem.