Lecture 3: The Heliocentric Model, Gravity, & Orbits

Lecture 3: The Heliocentric Model, Gravity, & Orbits Sec 3.3-3.6, 4.1-4.4 (4th Ed) Sec 3.2-3.5 (3rd Ed) As with all course material (including homewor...
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Lecture 3: The Heliocentric Model, Gravity, & Orbits Sec 3.3-3.6, 4.1-4.4 (4th Ed) Sec 3.2-3.5 (3rd Ed) As with all course material (including homework, exams), these lecture notes are not be reproduced, redistributed, or sold in any form. 

Where we left off… - Earth-centered model was favored due to lack of observed parallax (among other reasons) - Earth-centered model had difficult time reproducing retrograde motion of outer planets (e.g. Mars and Jupiter) - Aristotle’s/Ptolemy’s model of a perfect heavens with the Earth at the center of the Universe dominated (up to ~1600).

Tycho Brahe
 (1546-1601) •Had a metallic fake nose - lost in a math duel. •Made one of the most famous astronomical observations: a new star! (Nov 1572)

Tycho Brahe
 (1546-1601) •Had a metallic fake nose - lost in a math duel. •Made one of the most famous astronomical observations: a new star! (Nov 1572)

• For a few days, Tycho’s supernova was nearly as bright as the planet Venus. • Previously, people believed novas occurred in the Earth’s atmosphere. • Tyco showed the lack of parallax meant that it must be distant. • The perfection of Aristotle’s heavens was shattered.

Tycho Brahe
 (1546-1601)

• Soon famous for his work on this new star, King Frederick II of Denmark built Tycho a research center on the island of Hven between Denmark and Sweden. • Tycho made detailed observations of the motions of the stars and planets. – Did not use a telescope, rather a giant instrument called a “quadrant”. – Telescope was not yet invented (1608 in Netherlands).

Johannes Kepler
 (1571-1630)

• A gifted mathematician, who favored 


Copernicus’ heliocentric model. • Kepler wanted Tycho’s data to improve this model. • Tycho took Kepler on as an assistant, but guarded his data. • After Tycho’s death, Kepler inherited the data...

Johannes Kepler
 (1571-1630)

Kepler’s breakthrough...

• He considered whether the orbits of

the planets might be oval-shaped rather than perfect circles.

• In his model, the planets orbit the Sun on flattened circles (ellipses) -- a shape known for a long time (~300 BCE).

The Ellipse a

Every ellipse has two “Foci”.

b

The sum of the distances between every point on an ellipse and the Foci is always equal: a+b is the same all the way around.

minor axis major axis

Johannes Kepler
 (1609)

• Planets orbit in elliptical patterns, 


with the Sun at one focus.

• Deduced 3 laws to describe motion of (we’ll come back to these 3 laws).

planets 


Galileo Galilei
 (1564-1642)

• In the same year that Kepler published his work (1609) an Italian mathematician, Galileo Galilei, working 500 miles to the south, got wind of a new device called the telescope. • Galileo built and pointed a telescope towards the sky (a technological leap). • At the time, his telescope was the best (x10-30 power). Incredibly weak by today’s standards, but it changed everything. • Prior to Galileo, every astronomer in history had been studying the same stars and planets, just revising their movements and positions. • Galileo saw things never seen before: – New stars, structure on the moon, Jupiter had moons of its own.

Galileo Galilei
 (1564-1642)

With his telescope: 1] observed mountains on the moon 
 (again not all heavenly objects are perfect). 2] discovered the moons of Jupiter 
 (showing that not everything orbits around the Earth).

Galileo Galilei
 (1564-1642)

How Galileo finally killed the
 model of Ptolemy/Aristotle - In Ptolemy’s model, Venus & the Sun both orbit the Earth. - It had been known for thousands of years that Venus always appears in the sky very close to the Sun. -Therefore Ptolemy’s model forced the center of Venus’ epicycle to be in line with the Sun at all times.

Prediction: Venus will always appear as a “crescent” in the sky and never a “full” circle.

How Galileo finally killed the
 Ptolemy/Aristotle model - In the heliocentric theory, Venus orbits the Sun, and the Earth orbits the Sun, but at a larger distance. - The Earth’s orbit is always outside of Venus’. - Therefore Venus always appears close to the Sun in the sky, as has been known for thousands of years.

4) Venus will always appear Prediction as a “crescent” in the sky as a “crescent” Sometimes Venus will appear and as a “full circle” in the sky. andsometimes never a “full” circle.

How Galileo finally killed the
 Ptolemy/Aristotle model

Heliocentric Prediction

Ptolemaic Prediction

Phases of Venus Sun-centered Model

Earth-centered Model

Heliocentric theory: implications

• The Earth moves and rotates • The stars are very far away (why?) and thus the Universe itself is much bigger than people had imagined. • Aristotle was wrong. – The old ideas are not always right. – There is much to be learned. – People are capable of figuring out how the Universe works, even if it defies “common sense”.

What about the physics? Thanks to Copernicus, Tycho, Kepler, and Galileo, our new model consisted of planets orbiting the Sun along ellipses.

Johannes Kepler
 (1571-1630)

Kepler’s breakthrough...

• He considered whether the orbits of

the planets might be oval-shaped rather than perfect circles.

• In his model, the planets orbit the Sun on flattened circles (ellipses) -- a shape known for a long time (~300 BCE).

Kepler’s Laws 1. Law of Orbits: all planets move in elliptical orbits, 
 with the Sun at one focus. 2. Law of Areas: a line that connects a planet to the Sun
 sweeps out equal areas in equal times. 3. Law of Periods: the square of the period of any planet is 
 proportional to the cube of the semi-major axis of its orbit.

Kepler’s First Law

planets orbit the Sun along ellipses

planet

Focus

Sun

Eccentricity of an Ellipse b e= 1− − a

2

( ) √

Eccentricity of an Ellipse b e= 1− − a

2

( ) √

a

b

Eccentricity of an Ellipse b e= 1− − a

2

( ) √

a

b

eearth = 0.0167 esaturn = 0.0557 eneptune = 0.0112

Kepler’s Second Law a line that connects a planet to the Sun sweeps out equal areas in equal times

Slowly far from Sun

Quickly Near the Sun each open circle is spaced by equal amount of time

Kepler’s Second Law a line that connects a planet to the Sun sweeps out equal areas in equal times

Slowly far from Sun

Quickly Near the Sun each open circle is spaced by equal amount of time

Kepler’s Second Law a line that connects a planet to the Sun sweeps out equal areas in equal times

Slowly far from Sun

Quickly Near the Sun

Kepler’s Second Law a line that connects a planet to the Sun sweeps out equal areas in equal times

Slowly far from Sun

Quickly Near the Sun

Kepler’s Second Law a line that connects a planet to the Sun sweeps out equal areas in equal times

Slowly far from Sun

Quickly Near the Sun

Kepler’s Second Law a line that connects a planet to the Sun sweeps out equal areas in equal times

Kepler’s Third Law

the square of the period is proportional to the cube of the semi-major axis of the orbit. period = time it takes to go around once semi-major axis = something like the radius of the orbit

Kepler’s Third Law

the square of the period is proportional to the cube of the semi-major axis of the orbit.

A

2 P



3 A

Kepler’s Third Law

orbits of our planets are very close to circular

esaturn = 0.0557

A

2 P



3 A

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun?

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun? What we know: Pearth = 1 year Rearth = 1 AU = 1.5 × 1011 m (distance from Earth to Sun) Pjupiter = 11.85 years

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun? What we know: Pearth = 1 year Rearth = 1 AU = 1.5 × 1011 m (distance from Earth to Sun) Pjupiter = 11.85 years And from Kepler’s 3rd Law... P2 ∝ A3 where A = semi-major axis of elliptical orbit -or2 Pplanet 2 Pearth

=

3 Aplanet 3 Aearth

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun? 2 Pjupiter 2 Pearth

=

3 Ajupiter 3 Aearth

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun? 2 Pjupiter 2 Pearth

=

3 Ajupiter 3 Aearth

( )( ) Pjupiter 1 yr

2

=

Ajupiter 1 AU

3

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun? 2 Pjupiter 2 Pearth

=

3 Ajupiter 3 Aearth

( )( ) ( )( ) Pjupiter

2

1 yr

11.85 yr 1 yr

=

2

=

Ajupiter

3

1 AU

Ajupiter 1 AU

3

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun? 2 Pjupiter 2 Pearth

=

(

3 Ajupiter

140.42

3 Aearth

( )( ) ( )( ) Pjupiter

2

1 yr

11.85 yr 1 yr

=

2

=

Ajupiter

3

1 AU

Ajupiter 1 AU

3

) ( ) 1/3

=

Ajupiter AU

Kepler’s Third Law

an example: if the period of Jupiter is 11.85 years, 
 what is its distance from the Sun? 2 Pjupiter 2 Pearth

=

(

3 Ajupiter

140.42

3 Aearth

( )( ) ( )( ) Pjupiter

2

1 yr

11.85 yr 1 yr

=

2

=

Ajupiter

3

) ( ) 1/3

=

Ajupiter AU

Ajupiter = 5.2 AU

1 AU

Ajupiter 1 AU

3

Ajupiter = 5.2 AU = 7.8 × 1011 m

Example: Ratio Problem Assume that Jupiter is RJS = 5.2 AU away from the Sun. [Recall that 1 AU = 1.5 x 1011 m.] If we were to shrink the orbit of Jupiter down to 50m, how large would the Sun itself be? [On this scale, the whole orbit of Jupiter is about the size of a football field.]

Isaac Newton 
 (1642 - 1727) • Invented calculus (w/Leibniz) • Discovered gravity • Developed laws of motion 
 (foundation of modern physics) • Explained planetary motion (Kepler’s Laws) • Designed a new type of telescope • Expanded our knowledge of light • Also was philosopher, alchemist, etc.

The Time of Newton 
 (1642 - 1727)

• Most of the mathematical truths that had been discovered over history had been forgotten and then discovered, again and again, by multiple cultures. • It was possible for one person to understand nearly all of human knowledge --- only recently has this changed (~1800?). • Newton rediscovered most of mathematics and then invented calculus to explain motion. • Attended Cambridge, where the curriculum was based on the teachings of Aristotle (logic, cosmology, motion, etc.)

Aristotle’s View of Motion • Motion included any sort of change: pushing, pulling, spinning, etc. • A tendency of motion was dictated by the elemental composition of the object (air, water, fire, earth). • Light and heavy objects separated themselves naturally by moving up and down. • A more qualitative (less quantitative) picture of motion.

Influence of Galileo • Newton was aware of and influenced by Galileo. • Galileo concluded that all bodies fall at the same rate (acceleration not speed). • This conflicted with Aristotle’s “physics”.

‣ In 1664, Cambridge shut down because of the Plague. ‣ From home, Newton studied Euclid, Descartes, etc. ‣ Wanted to understand motion....to do this, he invented calculus (math that was continuous and not discrete).

Newton’s Calculus • Calculus = mathematical study of change - math that allows one to deal with infinitely large and small quantities. - describes how lines curve (their slope) and the areas enclosed by them.

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