Chapter%202:%20The%20Copernican%20Revolution%20The%20Birth%20of%20Modern%20Science

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Chapter 2 The Copernican RevolutionThe Birth of
Modern Science

• Ancient Astronomy
• Models of the Solar System
• Laws of Planetary Motion
• Newtons Laws
• Laws of Motion
• Law of Gravitation

2
Scientific Method

• Gather data

Form theory
Test theory
3
Astronomy in Ancient Times

• Ancient people had a better, clearer chance to
  study the sky and see the patterns of stars
  (constellations) than we do today.
• Drew pictures of constellations
  created stories to account for the figures being
  in the sky.
• Used stars and constellations for navigation.
• Noticed changes in Moons shape and position
  against the stars.
• Created accurate calendars of seasons.

4
Ancient Astronomy

• Stonehenge on the summer solstice.
• As seen from the center of the stone circle,
• the Sun rises directly over the "heel stone" on
  the longest day of the year.

The Big Horn Medicine Wheel in Wyoming, built by
the Plains Indians. Its spokes and rock piles
are aligned with the rising and setting of the
Sun and other stars.
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Astronomy in Early Americas

• Maya Indians developed written language and
  number system.
• Recorded motions of Sun, Moon, and planets --
  especially Venus.
• Fragments of astronomical observations recorded
  in picture books made of tree bark show that
  Mayans had learned to predict solar and lunar
  eclipses and the path of Venus.
• One Mayan calendar more accurate than those of
  Spanish.

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Ancient Contributions to Astronomy

• Egyptians
• recorded interval of floods on Nile
• every 365 days
• noted Sirius rose with Sun when floods due
• invented sundials to measure time of day from
  movement of the Sun.
• Babylonians
• first people to make detailed records of
  movements of Mercury, Venus, Mars, Jupiter,
  Saturn
• only planets visible until telescope

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Greek Astronomy

• Probably based on knowledge from Babylonians.
• Thales predicted eclipse of Sun that occurred in
  585 B.C.
• Around 550 B.C., Pythagoras noted that the
  Evening Star and Morning Star were really the
  same body (actually planet Venus).
• Some Greek astronomers thought the Earth might be
  in the shape of a ball and that moonlight was
  really reflected sunlight.

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Time Line

• Ancient Greeks
• Pythagoras 6th century B.C.
• Aristotle 348-322 B.C.
• Aristarchus 310-230 B.C.
• Hipparchus 130 B.C.
• Ptolemy A.D. 140

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Pythagorean Paradigm

• The Pythagorean Paradigm had three key points
  about the movements of celestial objects
• the planets, Sun, Moon and stars move in
  perfectly circular orbits
• the speed of the planets, Sun, Moon and stars in
  the circular orbits is perfectly uniform
• the Earth is at the exact center of the motion of
  the celestial bodies.

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Aristotles UniverseA Geocentric Model

• Aristotle proposed that
• the heavens were literally composed of
  concentric, crystalline spheres
• to which the celestial objects were attached
• and which rotated at different velocities,
• with the Earth at the center (geocentric).

The figure illustrates the ordering of the
spheres to which the Sun, Moon, and visible
planets were attached.
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Planetary Motion

• From Earth, planets appear to move wrt fixed
  stars and vary greatly in brightness.
• Most of the time, planets undergo direct motion -
  moving W to E relative to background stars.
  
• Occasionally, they change direction and
  temporarily undergo retrograde motion - motion
  from E to W -before looping back.

(retrograde-move)
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Planetary Motion Epicycles and
Deferents

• Retrograde motion was first explained as follows
  
• the planets were attached, not to the concentric
  spheres themselves, but to circles attached to
  the concentric spheres, as illustrated in the
  adjacent diagram.
• These circles were called "Epicycles",and the
  concentric spheres to which they were attached
  were termed the "Deferents".
• (epicycle-move)

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Epicycle/Deferent Modifications

• In actual models, the center of the epicycle
  moved with uniform circular motion, not around
  the center of the deferent, but around a point
  that was displaced by some distance from the
  center of the deferent.

This modification predicted planetary motions
that more closely matched the observed motions.
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Further Modifiations

• In practice, even this was not enough to account
  for the detailed motion of the planets on the
  celestial sphere!
• In more sophisticated epicycle models further
  "refinements" were introduced

In some cases, epicycles were themselves placed
on epicycles, as illustrated in the adjacent
figure. The full Ptolemaic model required 80
different circles!!
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Ptolemy

• 127-151 A.D. in Alexandria
• Accomplishments
• completion of a geocentric model of solar
  system that accurately predicts motions of
  planets by using combinations of regular circular
  motions
• invented latitude and longitude (gave
  coordinates for 8000 places)
• first to orient maps with NORTH at top
  and EAST at right
• developed magnitude system to describe brightness
  of stars that is still used today

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Aristarchus

• 310-230 B.C.
• Applied geometry to find
• distance to Moon
• Directly measure angular diameter
• Calculate linear diameter using lunar eclipse
• relative distances and sizes of the Sun and Moon
• ratio of distances to Sun and Moon by observing
  angle between the Sun and Moon at first or third
  quarter Moon.
• Proposed that the Sun is stationary and that the
  Earth orbits the Sun and spins on its own axis
  once a day.

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Hipparchus

• 190-125 B.C.
• Often called greatest astronomer of
  antiquity.
• Contributions to astronomy
• improved on Aristarchus method for calculating
  the distances to the Sun and Moon,
• improved determination of the length of the year,
• extensive observations and theories of motions of
  the Sun and Moon,
• earliest systematic catalog of brighter stars ,
• first estimate of precession shift in the vernal
  equinox.

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Time Line

• Ancient Greeks
• Pythagoras 6th century B.C.
• Aristotle 348-322 B.C.
• Aristarchus 310-230 B.C.
• Hipparchus 130 B.C.
• Ptolemy A.D. 140
• Dark Ages A.D. 5th - 10th century
• Arabs translated books, planets positions
• China 1054 A.D. supernova Crab Nebula

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Heliocentric Model - Copernicus

• In 1543, Copernicus proposed that the Sun,
  not the Earth, is the center of the solar system.
  
• Such a model is called a heliocentric system.
• Ordering of planets known to Copernicus in this
  new system is illustrated in the figure.
• Represents modern ordering of planets.
• (copernican-move)

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Stellar Parallax

• Stars should appear to change their position with
  the respect to the other background stars as the
  Earth moved about its orbit.
• In Copernicus day, no stellar parallax was
  observed, so the Copernican model was considered
  to be only a convenient calculation tool for
  planetary motion.
• In 1838, Friedrich Wilhelm Bessel succeeded in
  measuring the parallax of the nearby, faint star
  61 Cygni. ( penny at 4 miles)

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Time Line

• Ancient Greeks
  Pythagoras 6th century B.C.
  Aristotle 348-322
  B.C. Aristarchus 310-230
  B.C. Ptolemy
  A.D. 140
• Dark Ages A.D. 5th - 10th century
• Renaissance Copernicus
  (1473-1543) Tycho
  Brahe Kepler Galileo
  (1546-1601) (1571-1630)
  (1564-1642) Newton
  (1642-1727)

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Galileo Galilei

• Galileo used his telescope to show that Venus
  went through a complete set of phases, just like
  the Moon.
• This observation was among the most important in
  human history, for it provided the first
  conclusive observational proof that was
  consistent with the Copernican system but not the
  Ptolemaic system.

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Galileo and Jupiter

• Galileo observed 4 points of light that changed
  their positions with time around the planet
  Jupiter.
• He concluded that these were objects in orbit
  around Jupiter.
• Galileo called them the Medicea Siderea-the
  Medician Stars in honor of Cosimo II de'Medici,
  who had become Grand Duke of Tuscany in 1609.

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Time Line

• Ancient Greeks
  Pythagoras 6th century B.C.
  Aristotle 348-322
  B.C. Aristarchus 310-230
  B.C. Ptolemy
  A.D. 140
• Dark Ages A.D. 5th - 10th century
• Renaissance Copernicus
  (1473-1543) Tycho
  Brahe Kepler Galileo
  (1546-1601) (1571-1630)
  (1564-1642) Newton
  (1642-1727)

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Tycho Brahe
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Tycho Brahe

• Danish astronomer
• Studied a bright new star in sky that faded
  over time.
• In 1577, studied a comet
• in trying to determine its distance from Earth by
  observing from different locations, noted that
  there was no change in apparent position
• proposed comet must be farther from Earth than
  the Moon.
• Built instrument to measure positions of planets
  and stars to within one arc minute (1).

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Johannes KeplerLaws of Planetary Motion
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Kepler Elliptical orbits

• The amount of "flattening" of the ellipse is the
  eccentricity. In the following figure the
  ellipses become more eccentric from left to right.

A circle may be viewed as a special case of an
ellipse with zero eccentricity, while as the
ellipse becomes more flattened the eccentricity
approaches one.
(eccentricity-anim)
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Elliptical Orbits and Keplers Laws

• Some orbits in the Solar System cannot be
  approximated at all well by circles

- for example, Plutos separation from the Sun
varies by about 50 during its orbit!
According to Keplers First Law, closed orbits
arounda central object under gravity are
ellipses.
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As a planet moves in an elliptical orbit, the Sun
is at one focus (F or F) of the ellipse.
r
C
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The line that connects the planets point of
closest approachto the Sun, the perihelion ...
As a planet moves in an elliptical orbit, the Sun
is at one focus (F or F) of the ellipse
perihelion
v
r
C
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and its point of greatest separation from the
Sun, the aphelion
As a planet moves in an elliptical orbit, the Sun
is at one focus (F or F) of the ellipse
perihelion
is called the major axis of the ellipse.
v
r
C
aphelion
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The only other thing we need to know about
ellipses is howto identify the length of the
semi-major axis, because that determines the
period of the orbit.
Semi means half, and so the semi-major axis a
is half thelength of the major axis
v
r
C
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Keplers 1st Law

• The orbits of the planets are ellipses, with
  the Sun at one focus of the ellipse.

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Keplers 2nd Law

• The line joining the planet to the Sun sweeps out
  equal areas in equal times as the planet travels
  around the ellipse.

Orbit-anim
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An object in a highly elliptical orbit travels
very slowlywhen it is far out in the Solar
System,
but speeds up as it passes the Sun.
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According to Keplers Second Law,
the line joining the object and the Sun ...
38
sweeps out equal areas in equal intervals of
time.
equal areas
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That is, Keplers Second Law states that
The line joining a planet and the Sun sweeps
outequal areas in equal intervals of time.
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For circular orbits around one particular mass -
e.g. the Sun - we know that the period of the
orbit (the time for one completerevolution)
depended only on the radius r
- this is Keplers 3rd Law
M
For objects orbiting a common central body (e.g.
the Sun)in approximately circular orbits,
r
r
m
v
the orbital period squared is proportional to the
orbital radius cubed.
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Lets see what determines the period for an
elliptical orbit
For elliptical orbits,the period dependsnot on
r, but on thesemi-major axis a instead.
v
r
C
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It turns out that Keplers 3rd Law applies to
all ellipticalorbits, not just circles, if we
replace orbital radiusby semi major axis
For objects orbiting a common central body (e.g.
the Sun)
the orbital period squared is proportional to
the orbital radius cubed.
the orbital period squared is proportional to
the semi major axis cubed.
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So as all of these elliptical orbits have the
same semi-majoraxis a, so they have the same
period.
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So if each of these orbits is around the same
massiveobject (e.g. the Sun),
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So if each of these orbits is around the same
massiveobject (e.g. the Sun),
then as they all have the same semi-major axis
length a,
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So if each of these orbits is around the same
massiveobject (e.g. the Sun),
then as they all have the same semi-major axis
length a,
then, by KeplersThird Law, they have the
sameorbital period.
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Ellipses and Orbits

• Ellipse animation

48
Keplers 3rd Law

• The ratio of the squares of the revolution
  periods (P) for two planets is equal to the ratio
  of the cubes of their semi-major axes (a).

P2 a3 or P2/a3 1 where P is the
planets sidereal orbital period
(in Earth years) and a is the length of
the semi-major axis (in astronomical
units)
49
Astronomical Unit

• One astronomical unit is the semi-major
  axis of the Earths orbit around the Sun,
  essentially the average distance between Earth
  and the Sun.
• abbreviation A.U.
• one A.U. 150 x 106 km

50
Keplers 3rd Law for the Planets
P2 a3 or P2/a3 1
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Planetary Motions

• The planets orbits (except Mercury and Pluto)
  are nearly circular.
• The further a planet is from the Sun, the greater
  its orbital period.
• Although derived for the six innermost planets
  known at the time, Keplers Laws apply to all
  currently known planets.
• Do Keplers laws apply to comets orbiting
  the Sun?
• Do they apply to the moons of Jupiter?

52
Keplers Laws

• 1st Law Each planet moves around the Sun in an
  orbit that is an ellipse, with the Sun at one
  focus of the ellipse.
• 2nd Law The straight line joining a planet and
  the Sun sweeps out equal areas in equal intervals
  of time.
• 3rd Law The squares of the periods of
  revolution of the planets are in direct
  proportion to the cubes of the semi-major axes of
  their orbits.

53
Whats important so far?

• Through history, people have used the scientific
  method
• observe and gather data,
• form theory to explain observations and predict
  behavior
• test theorys predictions.
• Greeks produced first surviving, recorded models
  of universe
• geocentric (Earth at center of universe),
• other celestial objects in circular orbits about
  Earth, and
• move with constant speed in those orbits.
• Geocentric models require complicated
  combinations of deferents and epicycles to
  explain observed motion of planets. Ptolemaic
  model required 80 such combinations.
• Copernicus revived heliocentric model of solar
  system, but kept circular, constant speed orbits.

54
Whats important so far? continued

• Without use of a telescope, Tycho Brahe made very
  accurate measurements of the positions of
  celestial objects.
• Johannes Kepler inherited Brahes data and
  determined three empirical laws governing the
  motion of orbiting celestial objects.
• 1st Law Each planet moves around the Sun in an
  orbit that is an ellipse, with the Sun at
  one focus of the ellipse.
• 2nd Law The straight line joining a planet and
  the Sun sweeps out equal areas in equal
  intervals of time.
• 3rd Law The squares of the periods of revolution
  of the planets are in direct proportion to
  the cubes of the semi-major axes of their
  orbits.
• Galileo used a telescope to observe the Moon and
  planets. The observed phases of Venus validated
  the heliocentric model proposed by Copernicus.
  Also discovered 4 moons orbiting Jupiter,
  Saturns rings, named lunar surface features,
  studied sunspots, noted visible disk of planets
  (stars - point sources).

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Why do the planets move according to Keplers
laws? Or, more generally, why do objects move as
they do?
56
Historical Views of Motion

• Aristotle two types of motion
• natural motion
• violent motion
• Galileo
• discredited Aristotelian view of motion

Animations Air resistance Free-fall
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Galileo Why do objects move as they do?
speed increases.
speed decreases.
does speed change?
Without friction, NO, the speed is constant!
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What is a natural state of motion for an
object?
Moving with constant velocity?
At rest?
59
Inertia and Mass
Inertia a bodys resistance to
a change in its motion.
Mass a measure of an
objects inertia or, loosely, a measure of
the total amount of matter contained within an
object.
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Newtons First Law

• Called the law of inertia.
• Since time of Aristotle, it was assumed that a
  body required some continual action on it to
  remain in motion, unless that motion were a part
  of natural motion of object.
• Newtons first law simplifies concept of motion.

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Animation collision-1st-law
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FORCES and MOTION

• An object will remain
• (a) at rest or
• (b) moving in a straight line at constant speed
  until
• (c) some net external force acts on it.

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What if there is an outside influence?

• To answer this question, Newton invoked
  the concept of a FORCE acting on a body to cause
  a change in the motion of the body.

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Forces can act
through contact
instantaneously (baseball bat
making contact with the baseball),

or at a distance.
or continuously (gravity
keeping the baseball from flying into space).
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Velocity and Acceleration
Velocity describes the change in
position of a body divided by the time interval
over which that change occurs.
Velocity is a vector quantity, requiring
both the speed of the body and its direction.
Acceleration The rate of change of
the velocity of a body, any change in
the bodys velocity speeding up, slowing down,
changing direction.
Animation circularmotion
67
Newtons Second Law F ma

• Relates
• net external force F applied to object of
  mass m
• to resulting change in motion of object,
  acceleration a.

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If there is a NET FORCE on an object,how much
will the object accelerate?
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Newton and Gravitation

• Newtons three laws of motion enable calculation
  of the acceleration of a body and its motion,
  BUT must first calculate the forces.
• Celestial bodies do not touch ------ do not
  exert forces on each other directly.
• Newton proposed that celestial bodies exert an
  attractive force on each other at a distance,
  across empty space.
• He called this force gravitation.

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• Isaac Newton discovered that two bodies share a
  gravitational attraction, where the force of
  attraction depends on both their masses

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• Both bodies feel the same force, but in opposite
  directions.

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This is worth thinking about - for example, drop
a pen to the floor. Newtons laws say that the
force with which the pen is attracting the Earth
is equal and opposite to the force with which
the Earth is attracting the pen, even though the
pen is much lighter than the Earth!
75

• Newton also worked out that if you keep the
  masses of the two bodies constant, the force of
  gravitational attraction depends on the distance
  between their centers

mutual force of attraction
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• For any two particular masses, the gravitational
  force between them depends on their separation
  as

as the separation between the masses is
increased, the gravitational force of
attractionbetween them decreases quickly.
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Gravity and Weight

• The weight of an object is a measure of the
  gravitational force the object feels in the
  presence of another object.
• For example on Earth, two objects with different
  masses will have different weights.
• Fg m(GmEarth/rEarth2) mg
• What is the weight of the Earth on us?

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Mass and Weight

• Mass A measure of the total
  amount of matter contained within an object
  a measure of an objects
  inertia.
• Weight The force due to gravity
  on an object.
• Weight and mass are proportional.
• Fg mg where m
  mass of the object and g acceleration
  of gravity acting on the object

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Free Fall

• If the only force acting on an object is force of
  gravity (weight), object is said to be in a state
  of free fall.
• A heavier body is attracted to the Earth with
  more force than a light body.
• Does the heavier object free fall faster?
• NO, the acceleration of the body depends on both
• the force applied to it and
• the mass of the object, resisting the motion.
• g F/m F/m

82
Newtons Law of Gravitation

• We call the force which keeps the Moon in its
  orbit around the Earth gravity.

Sir Isaac Newtons conceptual leap in
understandingof the effects of gravity largely
involved his realizationthat the same force
governs the motion of a falling objecton Earth -
for example, an apple - and the motion of the
Moon in its orbit around the Earth.
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• Your pen dropping to the floor and a satellite in
  orbit around the Earth have something in common -
  they are both in freefall.

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Planets, Apples, and the Moon

• Some type of force must act on planet otherwise
  it would move in a straight line.
• Newton analyzed Keplers 2nd Law and saw that the
  Sun was the source of this force.
• From Keplers 3rd Law, Newton deduced that the
  force varied as 1/r2.
• The force must act through a distance, and
  Newton knew of such a force - the one that
  makes an apple accelerate downward from the tree
  to the Earth as the apple falls.
• Could this force extend all the way to the Moon?

85
To see this, lets review Newtons thought
experiment Is it possible to throw an object
into orbit around the Earth?
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On all these trajectories,the projectile is in
free fall under gravity.(If it were not, it
would travel in a straight line - thats
NewtonsFirst Law of Motion.)
87
If the ball is not given enough sideways
velocity, its trajectory intercepts the Earth
...
that is, it falls to Earth eventually.
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On the trajectories which make complete orbits,
the projectile is travelling sideways fast
enough ...
On all these trajectories, the projectile is in
free fall.
On all these trajectories, the projectile is in
free fall.
89
that as it falls, the Earth curves away
underneathit, and the projectile completes
entire orbits without ever hitting the Earth.
On all these trajectories, the projectile is in
free fall.
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Gravity and Orbits

• The Suns inward pull of gravity on the planet
  competes with the planets tendency to continue
  moving in a straight line.

91
Apparent Weightlessness in Orbit
This astronaut on a space walk is alsoin free
fall.
The astronauts sideways velocityis
sufficient to keephim or her in orbitaround the
Earth.
92
Lets take a little time to answer the following
question

• Why do astronauts in the Space Shuttle in Earth
  orbit feel weightless?

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• Some common misconceptions which become apparent
  in answers to this question are

(a) there is no gravity in space, (b) there is no
gravity outside the Earths atmosphere, or (c) at
the Shuttles altitude, the force of gravity is
very small.
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In spacecraft (like the Shuttle) in Earth orbit,
astronauts are in free fall, at the same rate as
their spaceships.
On all these trajectories, the projectile is in
free fall.
That is why they experience weightlessness just
as a platform diver feels while diving down
towards a pool, or a sky diver feels while in
free fall.
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Newtons Form of Keplers 3rd Law

• Newton generalized Keplers 3rd Law to include
  sum of masses of the two objects in orbit about
  each other (in terms of the mass of the Sun).
• (M1 M2) P2 a3
• Observe orbital period and separation of a
  planets satellite, can compute the mass of the
  planet.
• Observe size of a double stars orbit and its
  orbital period, deduce the masses of stars in
  binary system.
• Planet and Sun orbit the common center of mass of
  the two bodies.
• The Sun is not in precise center of orbit.

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Mass of Planets, Stars, and Galaxies

• By combining Newtons Laws of Motion and
  Gravitation Law, the masses of
  astronomical objects can be calculated.
• a v2/r , for circular orbit of radius r
• F ma mv2/r
• mv2/r Fg GMm/ r2
• v (GM/r)1/2
• P 2?r/v 2? (r3/GM)1/2
• M rv2/G
• If the distance to an object and the orbital
  period of the object are known, the mass can be
  calculated.