Chapter 3
Scales and Motion
in the Universe
Investigating Astronomy,
Slater & Freedman
In this chapter you will discover…
• What makes a theory scientific
• The scientific revolution that changed the idea of
an unmoving Earth and allowed the Earth to move
• Copernicus’s argument that the planets orbit the
Sun
• Why the direction of motion of the planets on the
celestial sphere sometimes appears to change
• That Kepler’s determination of the shapes of
planetary orbits depended on the careful
observations of his mentor Tycho Brahe
• How Isaac Newton formulated an equation to
describe the force of gravity and how he thereby
explained why the planets and moons remain in
orbit
The Ancient Mystery of the Planets
Our goals for learning:
• What was once so mysterious about the
movement of planets in our sky?
• Why did the ancient Greeks reject the real
explanation for planetary motion?
Planets Known in Ancient
Times
• Mercury
– difficult to see; always close
to Sun in sky
• Venus
– very bright when visible —
morning or evening “star”
• Mars
– noticeably red
• Jupiter
– very bright
• Saturn
– moderately bright
Eratosthenes and Aristarchus
Using simple tools and
basic geometry to
measure:
1. the size of the
Earth, Moon, and Sun
(310-~230 BC)
220? – 143? BC
2. the distances to the
Moon and Sun
1st heliocentric theory
Sun 18 to 20 x Moon
distance
Size of sun ~ 7x earth
What was once so mysterious
about planetary motion in our sky?
• Planets usually move eastward from night to
night relative to the stars.
– You cannot see this motion on a single
night; rather, planets rise in the east and
set in the west.
• But sometimes they go westward for a few
weeks or months: retrograde motion
Early models of the universe attempted to explain the motion
of the five visible planets against the background of “fixed”
stars. The main problem was that the planets do not move
uniformly against the background of stars, but at times appear
to stop, move backward, then move forward again. This
backward motion is referred to as retrograde motion.
Time-lapse images of Mars
during retrograde
Jupiter retrograde motion
The retrograde motion of Mars as
shown in a series of images taken
on the same photographic plate.
Explaining Apparent Retrograde
Motion
• Easy for us to explain: occurs when we
“lap” another planet (or when Mercury
or Venus lap us)
• But very difficult to explain if you think
the solar system is geocentric and the
Earth is unmoving
• In fact, ancients considered but
rejected the correct explanation…
We see apparent retrograde motion
when we pass by a planet in its orbit.
Why did the ancient Greeks reject the
real explanation for planetary motion?
• Their inability to observe stellar parallax was a major factor.
The Greeks knew that the lack of observable
parallax could mean one of two things:
1. Stars are so far away that stellar parallax is
too small to notice with the naked eye
2. Earth does not orbit Sun; it is at the bottom
of the universe
With rare exceptions such as Aristarchus, the
Greeks rejected the correct explanation (1)
because they did not think the stars could
be that far away…
Thus setting the stage for the long, historical showdown
between Earth-centered and Sun-centered
systems.
The most sophisticated geocentric model was that of Ptolemy
(A.D. 100-170) — the Ptolemaic model:
• Sufficiently accurate to
remain in use for 1,500
years.
• Arabic translation of
Ptolemy’s work named
Almagest (“the greatest
compilation”)
•Greeks also tended to
Believe that planets were
living beings influencing
man’s life..
Ptolemy
•Ptolemy’s book,
Tetrabiblios, is the bible of
astrology.
The Ancient Greek Model
An Earth-centered, or geocentric, model of
the universe
Ptolemy’s model used a geocentric (Earth-centered)
model of the solar system in which the planets
orbited the Earth indirectly by moving on epicycles
which in turn orbited the earth.
The Ptolemaic system was an ingenious and
complicated system of circular orbits
centered on other circular orbits called
epicycles. It remained the best model for
over 1500 years (with many modifications).
Earth Centered
Celestial sphere
The assumptions for this model were
The
commonly accepted:
1. the earth did not move
2. the earth was the center of the system
3. the stars were located at a fixed distance
on a transparent celestial sphere that
rotated from E to W
4. the celestial realm was unchanging, and
celestial motion was perfect, i.e. circular!
The Marriage of Aristotle and Christianity
•
•
In the 13th century St. Thomas Aquinas blended the
natural philosophy of Aristotle, which included the
Ptolemaic model, with Christian beliefs.
A central, unmoving Earth fit perfectly with prevalent
Christian thinking, and various scriptures where found,
whose literal interpretation, seemed to agree with this
model.
o
o
o
o
1 Chronicles 16:30: “He has fixed the earth firm, immovable.”
Psalm 96:10: “He has fixed the earth firm, immovable ...”
Psalm 104:5: “Thou didst fix the earth on its foundation so
that it never can be shaken.”
Isaiah 45:18: “...who made the earth and fashioned it, and
himself fixed it fast...”
Timeline of Ancient Astronomy
Ptolemy’s system worked well in general
detail. It was used to create tables predicting
the occurrence of astronomical events....
eclipses
conjunctions
etc.
Over several hundred years, small errors in the
tables accumulated to produce large error in
the timing of events - as much as a month by
1200 AD
A major revision was done in 1250 by a group
of scholars under King Alfonso of Spain.
Ptolemaic system was modified to include
deferents (off center circles).
They produced the Alfonsine Tables
By 1500 even these tables were in error by
several hours and even days in some cases
epicycle
Epicycle with deferent
center of
epicycle
Copernicus, a contemporary of
Columbus, worked 40 years on a
heliocentric—sun-centered—model for
two reasons:
(1) Ptolemy’s predicted positions
for celestial objects had
become less accurate over
time.
(2) The Ptolemaic model was not
aesthetically pleasing enough.
He wanted to restore perfect”
or circular motion and get rid
of off-center circles!
Because both models (Ptolemaic &
Copernican) were based on the assumption
that the planets move at constant speed,
Copernicus was forced to add small
epicycles of his own to improve accuracy.
Copernicus would not abandon the circle as
the preferred planetary orbit because he
thought circles are the best representation
of the perfect motions of the heavens.
Advantages of Copernican System:
Simpler! [not more accurate!]
simple explanation of retrograde motion
explained the phases of Venus
explained why Mercury & Venus always
close to the sun
Using trigonometry was able to
calculate the relative distances to
all visible planets
Timeline of Renaissance Astronomy
But What about the Scriptural
Evidence for the Geocentric Model?
• As more and more evidence began to build
which indicated the correctness of Copernicus’
model, faithful Christians had to ask some
fundamental questions about their interpretation
of scripture.
• By the end of the 17th century, most Christians
had come to accept the heliocentric model.
• These Christians had to make adjustments to
their interpretation of certain scriptures: the
Earth being “fixed” must be interpreted
differently.
The basic scriptural re-interpretation typically
involved asking the question, “What is the
scripture talking about in the verses interpreted
previously as a fixed Earth”?
The re-examination of scripture continues even
today as we seek the message of scripture that
God intended to deliver – which we have
discovered is almost never scientific information.
Comparing The Two Models
There were strong argument against
Copernican idea of a moving earth:
Inertia-if earth is moving, why don’t
objects thrown upward fall behind as
the earth rotates under the object?
Parallax-if earth moves, one should see
stellar parallaxes (stars seem to move
as viewed from different locations)
Celestial Sphere
Stellar parallax is quite small- 0.75
arcseconds for largest shift detected —
because the stars are so far away from us.
Stellar parallax, the apparent annual
shifting of nearby stars with respect to
background stars, was not observed until
1838.
The Copernican Model had good
Predictive Power
A good model (or theory) will make
verifiable predictions that might allow
the the theory to be disproved.
Using the Astronomical Unit (AU)—
the average distance between Earth and
Sun— Copernicus predicted with
amazing accuracy the Sun-to-planet
distances for the 5 planets visible from
Earth in the 1500s.
Planetary Distances in AU
Copernicus Value
Actual Value
Mercury
0.38
0.39
Venus
0.72
0.72
Earth
1.00
1.00
Mars
1.52
1.52
Jupiter
5.22
5.20
Saturn
9.17
9.54
The Copernican model was more
aesthetic since it could explain the
motions of Mercury and Venus without
resorting to special rules needed by the
Ptolemaic model.
Copernicus offered a simpler explanation
for retrograde motion that required no
use of epicycles.
16
Copernicus, who died in 1543 just as
his book De Revolutionibus was
published, started such an upheaval in
people’s thinking that the word
“revolution” took on a second meaning
that is so familiar to us today.
Tycho Brahe
Tycho was born 3 years after Copernicus died.
17
Tycho Brahe
Tycho built the largest and most accurate
naked-eye instruments yet constructed.
He could measure angles to within 0.1º,
close to the limit the human eye can
observe.
18
He not only made careful measurements, but
he recorded the accuracy of each
measurement.
1563 close conjunction of Jupiter & Saturn.
Alfonsine tables were off by a month, while
Copernican tables were off by several days.
Tycho wanted to correct the tables
19
Woodcut of Tycho
Silver nosepiece is
visible in the picture
Tycho lost the tip of his
nose in a duel at age 20
over a question in math.
11/11/1572 Nova appeared in
the sky and was closely
observed by Brahe: he
observed....
a. no apparent parallax, therefore the nova
was not inside celestial sphere
b. thus an obvious change in the unchanging
celestial sphere
Tycho writings about the Nova gained the
attention/approval of Frederick, king of
Denmark who built Tycho the world’s best
observatory (Uraniborg) on island of Ven
Tycho’s
supernova today
Woodcut of
Tycho’s Stella
Noveau
Woodcut of the inside of
Brahe’s observatory
Blaeu ´s Atlas , 1663
Stjerneborg, 1584, partly underground
Aerial view of site of Uraniborg on the Island Ven
Sweden
Denmark
Poland
After Frederick’s death, Tycho fell out of
favor and thus disassembled his
observatory and moved it to Prague under
HRE Rudolph II
Castle Benatky
His commission was to revise Alfonsine
tables
Hired several mathematicians to handle
the drudgery of the computations, one of
whom was Johannes Kepler
Tycho’s model
Tycho Brahe died 24th October 1601 of a urinary
bladder infection that he may have tried to cure
himself, with a medicine containing mercury
Teyn Church in Prague
where Tycho was buried
Body exhumed in 1901 to
determine cause of death
1996 Particle Induced X-ray Emission (PIXE)
showed recent high levels of Mercury in Brahe’s
hair samples implying mercury poisoning.
1610 painting by
unknown artist
Born 1571
Died 1630 (58)
Johannes Kepler
Kepler was a sickly child of a protestant family
living in predominantly catholic area.
Got scholarship to become Lutheran minister,
but liked math better. Had influential teacher
who was a Copernican.
Became Math teacher at Graz (not very good,
only had 1 student last year)
1595 wrote almanac with astronomical &
astrological weather predictions. They were
correct and got reputation as astrologer
In 1600, a year before Tycho died,
Kepler accepted a position as Tycho’s
assistant, working on calculations
Tycho’s best data had been gathered
for Mars.
Based on circles and epicycles Kepler’s
best Copernican model for Mars
matched Tycho’s data to within 0.13º
(8 arcminutes) [less than the accuracy
of Tycho’s measurements].
When Brahe died in 1601, Kepler got his job,
and after a fight with Brahe’s widow, got
possession of Brahe’s notebooks of data
The error in the position of Mars exceeded
the error in Tycho’s measurements, which
continued to bothered Kepler. Could get
agreement within 8 arcmin, > Tycho data
had a maximum error of 6 arcmin.
Kepler was lifelong mystic, enamored with
numbers (we would say a numerologist)
In possession of Brahe’s data,
Kepler spent more than 5 years
pouring over the details, trying to
reconcile the error.
Kepler’s persistence finally led him
to abandon circles and try other
shapes. The shape that worked
for Mars and all other planets was
the ellipse.
The Ellipse
The ellipse is a geometrical
shape every point of which is the
same total distance from two fixed
points (the foci).
Eccentricity is the distance
between the foci and its center
divided by half the longest
distance across (semi-major axis).
sun
focus
The eccentricity of the ellipse
measures the difference
between the major and minor
axes. e = c/a
If the axes are equal,
then e=0 and the
ellipse becomes a
circle.
minor axis
major axis
c
planet
focus
a
All the planet orbits have
e ~ 0.1 except Pluto (.248)
and Mercury (.206)
The center of force occupies one focus of the
ellipse, while the other focus is usually empty
Kepler also discovered what we call the
Law of Equal Areas which showed that
planets did NOT move at constant
speeds in their orbits
Kepler’s 2nd Law – the law of equal areas
1 month
1 month
After more than 10 years further work, Kepler
wrote a rather obscure and mystical book that
showed a relationship between a planet’s orbit
radius (a in AU) and its orbital period (P in years)
P2 = a3
All of his discoveries are called Kepler’s 3 Laws of
Motion
In addition to the Laws of Motion, Kepler is
also one of the 1st to try to give a physical
reason for planets orbiting the sun. He
thought that some type of magnetic force
was responsible
Galileo Galilei
Born in Italy (1564), a
contemporary of Copernicus. He
was a Prof. at Padua in the
Venetian Republic, & a Prof. at the
University of Florence
Strong believer in experimentation
Strong, abrasive personality,
popular writer who wrote in
common Italian rather than Latin.
He was very free to criticize and
ridicule any who differed with him
on any matter.
Galileo Galilei and the Telescope
• Galileo built his first telescope in 1609,
shortly after hearing about telescopes
being constructed in the Netherlands.
• Galileo was perhaps the first person to use
a telescope to systematically study the sky
and record his observations.
1
Galileo made 5 important
observations:
Mountains and valleys on the Moon
Sunspots
More stars than can be observed
with the naked eye
Four moons orbiting Jupiter
Complete cycle of phases of Venus
• Though Galileo’s first three observations
do not disprove the geocentric theory,
they cast doubt on the the assumption
of perfection in the heavens.
The existence of stars too dim to be
seen with the naked eye also cast
doubt on the the fact that stars were
all the same.
In 1610, Galileo
discovered four “stars”
that move back and
forth across Jupiter. He
concluded that they are
four moons that orbit
Jupiter just as our Moon
orbits Earth. These
observations made
by Jesuits in 1620 of
Jupiter and its four
visible moons.
Galileo observed that Venus goes
through a full set of phases: full,
gibbous, quarter, crescent.
Venus’s full set of phases can be
explained by the heliocentric theory.
The Ptolemaic theory predicts that
Venus will always appear in a crescent
phase, which is not borne out by the
observations.
Venus’ Phases in Ptolemy’s Model
Galileo is also the first person to do any
systematic study of motion and was the first
person to understand the concept of inertia
Galileo did a wide variety of experiments,
even attempting to measure the speed of
light by using lanterns on distant hilltops
(concl. either infinite or too large to
measure)
Isaac Newton
• Galileo is credited with setting the
standard for studying nature through
reliance on observation and
experimentation to test hypotheses.
• The year Galileo died—1642—is the
year Isaac Newton was born.
• Newton took the work of Galileo and
Kepler and created an expansive
theory of motion.
6
Isaac Newton (1642–1727)
Isaac Newton was
undeniably one of
the greatest/most
influential scientists
that ever lived.
Very religious man
who believed the
order in the
universe was
representative of
God
Very cautious person who had to be
persuaded by his friends to publish ANY of his
work
• 1665-67 plague in London, Newton left
the university and went to his country
home for a year. While there he:
– developed basic ideas of mechanics
– basic concepts of gravity
– beginning ideas on light and optics
1687 published Principia ( perhaps the
most influential scientific book ever
published)
In it he explained the motion of the
planets, comets using his law of
gravity and his 3 Laws of Motion. He
also derived Kepler’s Laws
Could not solve the problem of gravitational
attraction on a planet so he invented Calculus
to solve it
Published a book Optiks giving the basic
ideas of geometrical optics, light, and color,
including the fact that white light is made up
of different colors
To understand the motion of objects under the
influence of gravity, we use the ideas of Isaac
Newton and his 3 Laws of motion.
The first concept needed is the idea of INERTIA.
This was the concept not understood by ancient
observers in their arguments against a moving
earth.....
“toss object in the air, if the earth is
moving then the object will fall behind its launch
point...”
While not the first person to properly conceive of
inertia, Galileo was the first to arrive at his views
based on actual experiments he performed.
Galileo’s work provided the basis for Newton’s
formulation of the Law of Inertia.
Inertia is the property of an object
whereby it tends to maintain whatever
velocity it has. The inertia of an object is
determined by it MASS.
• Newton’s First Law (Law of Inertia):
Unless an object is acted upon by a net,
outside force, the object will maintain a
constant speed in a straight line.
Note: a speed of zero (rest) is a
constant speed.
7
Block continues to move when the cart suddenly stops due to
the inertia of the block!
Newton’s Second Law says that the
acceleration of an object depends on
the force applied to it and on its mass!
• Acceleration is inversely proportional to
the mass being accelerated.
• What does “inversely proportional” mean?
– As the mass gets bigger, the acceleration
gets smaller
10
If an objects speed or direction of motion changes
(like the block) - we say that the object is
accelerated!
• Acceleration is a measure of how
rapidly the speed or direction of motion
of an object is changing.
• An object at rest has a speed of zero.
• Newton’s first law says that a force is
needed to change the speed and/or
direction of an object’s motion.
8
V=0
No force means no acceleration!
Car remains at rest (law of inertia)
V>0
F
The longer the force
acts, the longer the
car accelerates and
the faster it goes
If an unbalanced force is
applied, the car accelerates
and its speed increases >0
V>>0
F
V>>>0
F
An Important Digression —
Mass & Weight
Mass is the quantity of inertia an object
has. Produced by particles from which it
is made
• Mass is NOT volume or weight.
• Weight is the force of gravity.
• The international (SI) unit of mass is the
kilogram.
• A kilogram on Earth weighs about 2.2
pounds.
9
Acceleration = force divided by mass
In symbols
A = F/m
Often as Force = mass X acceleration
or F = mA
If the left side = 0 then right side is
also = 0
Thus when the net force is zero,
there is no acceleration.
A
=
F/m
if the force is large then Acceleration is
large:
A
=
F
m
if the mass is large then acceleration is
small
A = F
m
Newton’s Third Law
• Third Law: When object X exerts a force
on object Y, then object Y exerts an equal
and opposite force back on X.
X
Y
X pushes on Y
Y pushes on X
X
Y
The forces are equal in size and opposite in
direction
11
The Third Law is sometimes stated as
“For every action there is an opposite
and equal reaction,” but the first
statement is more precise in terms of
physical forces.
REMEMBER: The two forces ALWAYS act
on DIFFERENT objects.
Also there is no such thing as a single force!!
Motion in a Circle
• Motion of an object in a circle at
constant speed (uniform circular motion)
is an example of acceleration by
changing direction.
• Centripetal (“center-seeking”) force
is the force directed toward the center of
the curve along which the object is
moving.
13
The most common force to discuss
while studying the motion of
planets, comets, stars, galaxies,
and other such objects (including
balls, etc. on the Earth) is
GRAVITY.
As the Earth orbits the sun, the force of gravitational
attraction from the Sun pulls on the Earth
Velocity at a
and velocity at
b are NOT the
same – have
diff. directions
a
velocity
b
velocity
The Earth is accelerated continuously in its orbit!
The Law of Universal Gravitation
• This law states that between every two
objects there is an attractive force, the
magnitude of which is directly
proportional to the mass of each object
and inversely proportional to the
square of the distance between the
centers of the objects.
14
In equation form:
GM2 M 1
F
2
D
G is a constant, m1 and m2 are the masses,
and d is the distance between their centers.
Two objects that have mass are attracted
to each other.
M1
M2
F1
F2
D
F1 & F2 are equal except for direction!
Weight is the gravitational force between an
object and the planetary body on which the object
is located.
mass
Weight : pull of
planet on object
Earth
mass
W = mg
Pull of gravity on a mass on
Earth – known as weight
Points approx. toward the
center of the Earth or what
we call DOWN
–According to Newton, gravity….
--- makes objects fall to Earth
--- keeps the Moon in orbit around the Earth ---- keeps the planets in orbit around the Sun
–He could therefore explain the planets’ motions
and why Kepler’s laws worked.
The pull of the Earth’s gravity could be tied
in with the orbit of satellites (such as the
Moon) around the Earth.
Newton made an argument to show this that
is now known as Newton’s Cannon!
Applet
Newton’s Laws and Kepler’s Laws
• Kepler’s first law—the planets move in
elliptical orbits—can be derived from
Newton’s laws but requires calculus.
• Kepler’s second law—planets sweep out
equal areas in equal times—can also be
derived from Newton’s laws. As planets
orbit the Sun they show a change in both
speed and direction.
20
Newton and Kepler’s 3rd Law
• Newton showed mathematically that
Kepler’s third law—the perioddistance relationship—derives from
the inverse square law for gravitation.
• Newton modified Kepler’s third law,
showing that mass is an important
factor.
a3
p2
=kM
Where a is in meters
and p is in seconds
k is a constant and
M is mass in kilograms
21
The Center of Mass
Seesaw principle
• Center of mass is the average location
of the various masses in a system,
weighted according to how far each is
from that point. The CM is sometimes
called the center of gravity.
• Barycenter is the center of mass of
two astronomical objects revolving
around one another.
22
•The barycenter for the Earth-Moon system
is inside the Earth, 4641 km from its center
and inside its 6378 km radius
• The location of the center of mass of the
Earth-Moon system was determined by
observing parallax of nearby planets due
to the Earth’s motion as the Moon went
around.
23
Using Newton’s Laws of Motion allows us
to understand the general features of
satellite motion, such as the moon, or
any other orbiting satellite.
Careful measurements of the orbits
and periods of satellites (natural and
man-made) allow one to accurately
determine the mass of the body being
orbited.
One of the crowning achievements of
Newton’s gravitational law was the
discovery of Neptune.
After it’s accidental discovery, the orbit
of Uranus was analyzed using Newton’s
Laws and the Law of Gravity.
Despite careful measurements, the observed
orbit did not match the one predicted by
Newton’s Laws.
Uranus & 3 moons
The discrepancies were attributed to another
unknown mass orbiting outside Uranus’ orbit.
Using these results, the mass and location of the
unknown object was predicted.
Neptune & 1 moon
Assuming Newton was correct, the discrepancies
could only be explained by another planet, about
the same size, orbiting outside of the orbit of
Uranus.
Gravity Works at
All Scales
This figure
shows a few of
the effects of
gravity here on
Earth, in the
solar system, in
our Milky Way
Galaxy, and
beyond.
The Importance of Newton’s Laws
• Kepler’s laws can be derived from them.
• They explain tides and precession.
• Their use predicted the existence of the
planet Neptune.
• They provide a way to measure things
quantitatively and predict the motion of
things.
• Newton laid the foundation for our
concept of the Universe.
32
Beyond Newton: How Science Progresses
• Newton proposed that inertial mass was
equivalent to gravitational mass, but he
had no idea why. Subsequent
measurements confirmed this
coincidence.
• Einstein in his General Theory of
Relativity showed mathematically that
the two types of masses are indeed
equivalent.
33
This coincidence was one of the seeds
leading Einstein to the development of the
General Theory of Relativity
The End
Next, Chpt. 4
Exploring Our
Evolving Solar
System