Chapter 2 The Copernican Revolution The Birth of

Modern Science

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

Scientific Method

- Gather data

Form theory

Test theory

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.

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.

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.

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

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.

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

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.

Aristotles Universe A 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.

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)

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)

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.

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!!

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

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.

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.

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

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)

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)

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)

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.

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.

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)

Tycho Brahe

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).

Johannes Kepler Laws of Planetary Motion

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)

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

around a central object under gravity are

ellipses.

As a planet moves in an elliptical orbit, the Sun

is at one focus (F or F) of the ellipse.

r

C

The line that connects the planets point of

closest approach to 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

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

The only other thing we need to know about

ellipses is how to 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 the length of the major axis

v

r

C

Keplers 1st Law

- The orbits of the planets are ellipses, with

the Sun at one focus of the ellipse.

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

An object in a highly elliptical orbit travels

very slowly when it is far out in the Solar

System,

but speeds up as it passes the Sun.

According to Keplers Second Law,

the line joining the object and the Sun ...

sweeps out equal areas in equal intervals of

time.

equal areas

That is, Keplers Second Law states that

The line joining a planet and the Sun sweeps

out equal areas in equal intervals of time.

For circular orbits around one particular mass -

e.g. the Sun - we know that the period of the

orbit (the time for one complete revolution)

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.

Lets see what determines the period for an

elliptical orbit

For elliptical orbits, the period depends not on

r, but on the semi-major axis a instead.

v

r

C

It turns out that Keplers 3rd Law applies to

all elliptical orbits, not just circles, if we

replace orbital radius by 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.

So as all of these elliptical orbits have the

same semi-major axis a, so they have the same

period.

So if each of these orbits is around the same

massive object (e.g. the Sun),

So if each of these orbits is around the same

massive object (e.g. the Sun),

then as they all have the same semi-major axis

length a,

So if each of these orbits is around the same

massive object (e.g. the Sun),

then as they all have the same semi-major axis

length a,

then, by Keplers Third Law, they have the

same orbital period.

Ellipses and Orbits

- Ellipse animation

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)

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

Keplers 3rd Law for the Planets

P2 a3 or P2/a3 1

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?

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.

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.

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).

Why do the planets move according to Keplers

laws? Or, more generally, why do objects move as

they do?

Historical Views of Motion

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

Animations Air resistance Free-fall

Galileo Why do objects move as they do?

speed increases.

speed decreases.

does speed change?

Without friction, NO, the speed is constant!

What is a natural state of motion for an

object?

Moving with constant velocity?

At rest?

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.

(No Transcript)

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.

Animation collision-1st-law

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.

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.

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).

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

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.

(No Transcript)

If there is a NET FORCE on an object, how much

will the object accelerate?

(No Transcript)

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.

- Isaac Newton discovered that two bodies share a

gravitational attraction, where the force of

attraction depends on both their masses

- Both bodies feel the same force, but in opposite

directions.

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!

- 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

- 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

attraction between them decreases quickly.

(No Transcript)

(No Transcript)

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?

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

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

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

understanding of the effects of gravity largely

involved his realization that the same force

governs the motion of a falling object on Earth -

for example, an apple - and the motion of the

Moon in its orbit around the Earth.

- Your pen dropping to the floor and a satellite in

orbit around the Earth have something in common -

they are both in freefall.

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?

To see this, lets review Newtons thought

experiment Is it possible to throw an object

into orbit around the Earth?

On all these trajectories, the projectile is in

free fall under gravity. (If it were not, it

would travel in a straight line - thats

Newtons First Law of Motion.)

If the ball is not given enough sideways

velocity, its trajectory intercepts the Earth

...

that is, it falls to Earth eventually.

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.

that as it falls, the Earth curves away

underneath it, and the projectile completes

entire orbits without ever hitting the Earth.

On all these trajectories, the projectile is in

free fall.

Gravity and Orbits

- The Suns inward pull of gravity on the planet

competes with the planets tendency to continue

moving in a straight line.

Apparent Weightlessness in Orbit

This astronaut on a space walk is also in free

fall.

The astronauts sideways velocity is

sufficient to keep him or her in orbit around the

Earth.

Lets take a little time to answer the following

question

- Why do astronauts in the Space Shuttle in Earth

orbit feel weightless?

- 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.

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.

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.

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.