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Phys 181-701

Astronomy

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“The danger to which the success of revolutions is most exposed, is that of attempting them before the principles on which they proceed, and the advantages to result from them,

are sufficiently seen and understood.” Thomas Paine - American Revolutionary

“Anyone in a free society where the laws are unjust has an obligation to break the law.”

Henry David Thoreau - American Philosopher

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“Numero pondere et mensura Deus omnia condidit.” Sir Isaac Newton – Principia Mathematica

“If I have been able to see further, it was only because I stood on the shoulders of giants.” Newton, in a letter to Robert Hooke

“To command the professors of astronomy to confute their own observations is to enjoin an impossibility, for it is to

command them to not see what they do see, and not to understand what they do understand, and to find what they

do not discover.”Galileo Galilei – In Science

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(rěv΄ə-lōō’shən))1. n A drastic and far-reaching

change in ways of thinking and behaving.

2. n Orbital motion about a point, the planetary revolution about the sun.

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The Copernican Revolution

1473 - 1542

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Problems with Ptolemy

There are problems with the Ptolemaic Model

The solar system according to the Ptolemaic Model from 100 A.D. to 1500

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Features

Features…•Deferent Circles•Epicycles on the Deferents•Inferior planets are never more than 47º from the sun•Retrograde motion of mars is explained

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• As the precision of astronomical measurements grew the need for adjustments to the model called for additional epicycles.

• In 1252 King Alfonso X of Castile funded a 10 year project by Arab and Jewish astronomers to calculate extensive tables.

• Calculations became horrendously complicated and even Alfonso suggested that the model should by simpler if it was to have any truth in it.

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“Multiplicity ought not be posited without necessity.”

William Occam – English Scholar - 1340

“Keep it simple, stupid.”Anonymous 20th Century Philosopher

This principle is known as Occam’s Razor…

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Copernican Theory…

The motions of the planets would be more easily explained if the sun were at the center.

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DeRevolutionibus, 1543…

“Venus and Mercury revolve around the sun and cannot go nearer or further from it than the circles of their orbits permit…If, acting upon this supposition, we connect Saturn, Jupiter and Mars with the same center, keeping in mind the greater extent of their orbits…we cannot fail to see the explanation of the regular order of their motions.

This proves sufficiently that their center belongs to the sun…”

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Going Further…

“The extent of the universe…is so great that…[the earth] disappears when compared to the sphere of the fixed stars.”

“Although this may appear incomprehensible and contrary to the opinion of many, I shall, if God wills, make it clearer than the sun, at least to those who are not ignorant of mathematics.”

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“…that fool who would reverse the entire art of astronomy…Joshua bade the SUN and not the earth to stand still.”

Martin Luther - 1539

Under very real threat of persecution, Copernicus changed the title of his book and its publication was finished only in the final days of his life.

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Tycho Brahe•Royal Astronomer of Denmark

•Uraniborg

•Observations with the naked eye

•Measured parallax of planets

•Observed a supernova (1572)

•Realized that Ptolemy was not right but denied Copernicus

•Proposed a model with the earth at the center and the sun revolving around it, but all of the other planets revolved around the sun

•Collected the best data available

1546-1601

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KEPLER!

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KeplerJohannes Kepler – 1571-1630

Deeply religious believer in astrology

Mathematics was evidence of God

Subscriber to the Copernican theory

Hired by Tycho to prove the earth centered model

Convinced that a mathematical “harmony” existed for the planets

Inherited Tycho’s fine data

Analyzed data regarding mars

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Kepler’s laws

Kepler’s Laws

1. Each planet moves in an ellipse, with the sun at one focus.

2. The line between the sun and the planet sweeps out equal areas in equal times.

3. The ratio of the cube of the average radius of a planets orbit to the square of its orbital period of revolution is the same for each planet. (Harmonic Law)

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Ellipses and AreasThis orbit is in the shape of an ellipse….

The area AAB= The Area ACD

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Law of Conservation ofAngular Momentum

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Harmonic Law

The Harmonic Law:

kP

a

2

3

ka

P

3

2

a = Average radius of planet’s orbitP = The orbital period of the planetk = A constant for all objects orbiting the sun

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Harmonic ConstantConsider the earth…

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1

1

12

3

2

3

yr

AU

P

a

This is true for all planets no matter what the radius of their orbit.

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Retrograde MotionRetrograde motion explained at long last…

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How does Kepler know times?

How does Kepler know the time periods of planets other than the earth?

Consider…

•You can’t go to mars

•You can’t watch mars ‘go round’

•Every so often, mars exhibits retrograde motion

•Retrograde motion is caused by the passing of mars by the earth at various points in their orbits.

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Working late on this lecture, the clock strikes midnight…And the hands line up.

A little while later… And they aren’t lined up at all.

I notice that they line up again sometime around1:05 am. Is there some sort of a math rule that can predict when the hands will align?

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A little scratch work and some calculations…

Big deal…

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Time period of marsIf I use the hands of the clock as an analogy for the motion of the planets…

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Radius of mars’ orbit

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Measurement

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IMPORTANT

• Occam’s Razor• Copernicus• Law of Ellipses • Law of Areas• Harmonic Law

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Review

REVIEW:

Kepler Develops Three Laws:

•Law of Ellipses

•Law of Areas

•Harmonic Law

P

ak

2

3 constant

We now understand HOW the planets move… but not WHY they move.

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Galileo: The Death of the Earth Centered Universe

1564-1642

•Contemporary of Kepler

•Demonstrated that all objects are accelerated by gravity by the same amount

•Moving objects remain in motion

•Built a telescope in 1609* and observed the Sun, Moon, Milky Way, Moons of Jupiter and the phases of Venus.

*Hans Lippershey invented the telescope in 1608

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If we assume (incorrectly) that the Tower of Pisa is 20m tall, the ball will take 2s to hit the ground.

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Even if the ball is thrown horizontally from the tower, the acceleration toward the earth is still 10m/s2.

As a result, the ball that is dropped and the ball that is thrown both hit the ground after 2 seconds!!!

We will return to this essential idea in a few slides…

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Sir Isaac Newton

Newton’s Laws:

1. All objects at rest shall remain at rest and all objects in motion shall remain in motion in a straight line, unless compelled by a FORCE to do otherwise.

2. The ACCELERATION of any object is directly proportional to the FORCE applied to it and inversely proportional to its MASS.

3. For every force applied to an object, there is an equal and opposite force applied by the object on the actor.

1642-1727

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Newton’s Laws Relative to Galileo’s Experiment:

1. When the ball is dropped it ceases to be at rest. Therefore there must be a force, directed downward, to cause the acceleration.

2. The acceleration will be equal to the force that gravity exerts on the ball divided by the mass of the ball, that is, the acceleration is equal to the force per unit mass.

3. If the Earth exerts a gravitational force on the ball, the ball must exert an equal and opposite force on the Earth!!!!

Newton v.s. Galileo

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Law of Universal Gravitation

Newton knows that the more mass an object has, the greater the force of Gravity on it.

FG= m g

Where “g” is the special name given to the acceleration that is caused by gravity. 10 m/s2

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Inverse squareThe inverse square law…

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The Law

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Example

Example:

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Mass of baby

Mass of doctor

Distance between baby and doctor

Assumptions

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“Knowns”

Mass of Mars

Mars-Baby Distance

Universal Gravitational Constant

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“Weighing” the Earth…

A & B have equal masses and therefore equal weights.The rod is balanced.

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The very small mass is needed to balance the gravitational force of

the very large mass. “G” can be calculated!

Knowing G and Kepler’s Law’s allows us to calculate the mass of the Earth, Sun and all of the planets moons and asteroids in the solar system

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Return to Pisa

Return to Pisa…The earth is not flat…

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Curved Earth

After one second the projectile has fallen five meters…But the earth has curved away.

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Question

kP

a

2

3

Question…

If the earth is curved such that it “curves away” 5 meters for every 8000 meters traveled, how fast would the projectile need to be going so that, after falling 5 meters, it was still 5 meters above the earth?

8000 m/s!!!

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Summary•Universal Gravitation used to determine the mass of the Earth•Satellite motion possible•Solar system travel made possible

•Newton invents calculus•Newton Proves Kepler’s Laws•Tides understood•Moon “lock” understood

WE NOW UNDERSTAND…

WE ARE UNCLEAR ON…

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Calculus and Planetary Motion…

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R

R

v t

h

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WHICH IS NOT THE RIGHT ANSWER….

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This does not represent the true motion….

The true motion is revealed when weMake the time very, very, very small…

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Newton Tells us that…

An that, for gravity in particular…

We have just discovered that…

We may deduce then that…

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Determine the mass of the SUN….

R = 1.496 x 1011 m

T = 3.156 x 107 s

G = 6.673 x 10-11 Nm2/kg2

M = 1.99 x 1030 kg

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Forces are Balanced on a Spherical Moon

Forces in Competition on a Prolate Moon

Forces are Balanced when Collinear on a Prolate Moon

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Summary II

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IMPORTANT

• Objects fall at the same rate.• Newton’s Laws• Inverse Square Law of Gravity• Nature of Orbits• Renaissance Astronomers• Occam’s Razor• Copernicus• Law of Ellipses • Law of Areas• Harmonic Law

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IMPORTANT

• Occam’s Razor• Copernicus• Law of Ellipses • Law of Areas• Harmonic Law

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Electromagnetic Radiation

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