Astro110-01 Lecture 8
The Copernican Revolution
(Cont’d)
or the revolutionaries:
Nicolas Copernicus (1473-1543)
Tycho Brahe (1546-1601)
Johannes Kepler (1571-1630)
Galileo Galilei (1564-1642)
Isaac Newton (1642-1727)
who toppled Aristotle’s cosmos
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Johannes Kepler
(1571–1630)
• In the interplay between quantitative
observation and theoretical construction that
characterizes the development of modern
science, Brahe was the master of the first but
was deficient in the second.
• The next great development in the history of
astronomy was the theoretical intuition of
Johannes Kepler (1571-1630), a German who
went to Prague to become Brahe's assistant.
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Kepler and the Elliptical Orbits
• Unlike Brahe, Kepler believed firmly in the
Copernican system.
• Kepler realized that the orbits of the planets were
not the circles but were instead the "flattened
circles" called ellipses
The difficulties with the Martian orbit derive
precisely from the fact that the orbit of Mars was
the most elliptical of the planets for which Brahe
had extensive data.
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What is an ellipse?
An ellipse looks like an elongated circle.
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Eccentricity of an Ellipse
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Eccentricity and Semimajor Axis of an Ellipse
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Kepler’s three laws of planetary
motions
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Kepler’s First Law:
The orbit of each planet around the Sun is an
ellipse with the Sun at one focus.
[Greek: near
the Sun]
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[Greek:
away from
the Sun]
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Kepler’s Second Law:
As a planet moves around its orbit, it sweeps
out equal areas in equal times.
 A planet travels faster when it is nearer to the
Sun and slower when it is farther from the Sun.
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Kepler's 2nd Law
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Kepler’s Third Law
•
The ratio of the squares of the revolutionary periods for two
planets is equal to the ratio of the cubes of their semimajor axes:
•
Choosing subscript 1 for the Earth, the relation can be rewritten
as:
p2 = a3
with p = orbital period in years
and a = average distance from Sun in AU
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Kepler’s Third Law
 Kepler's Third Law implies that the period for
a planet to orbit the Sun increases rapidly with
the radius of its orbit.
 More distant planets orbit the Sun more
slowly than the ones that are closer
- Mercury, the innermost planet, takes only
88 days to orbit the Sun
- the outermost planet (Pluto) requires 248
years to do the same.
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Planets’ period
http://www.ac.wwu.edu/~stephan/Astronomy/planets.html
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Kepler’s Third Law (Cont’d)
• The only thing that affects the orbital
period p of the planets is the semimajor
axis a
• The mass, and orbital eccentricity, do
not matter
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Kepler’s Third Law
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Graphical version of Kepler’s Third Law
This graph shows that Kepler’s 3rd law
hold true. The graph shows the
planets that were known during
Kepler’s time
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This graph shows how the
orbital speeds of the planets
depend on their distances from
the Sun: More distant planets
orbit the Sun more slowly.
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Astro 110-01 Lecture 8
Clicker Question
An asteroid orbits the Sun at an average distance
a = 4 AU. How long does it take to orbit the Sun?
A.
B.
C.
D.
4 years
8 years
16 years
64 years
(Hint: Remember that p2 = a3.)
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Clicker Question
An asteroid orbits the Sun at an average distance
a = 4 AU. How long does it take to orbit the Sun?
A.
B.
C.
D.
4 years
8 years
16 years
64 years
We need to find p so that p2 = a3.
Since a = 4, a3 = 43 = 64.
Therefore p = 8, p2 = 82 = 64.
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Clicker question: Planetary orbits
When we say that a planet has a highly
eccentric orbit, we mean that:
1. it is spiraling in toward the Sun.
2. its orbit is an ellipse with the Sun at
one focus.
3. in some parts of its orbit it is much
closer to the Sun than in other parts.
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Clicker question: Planetary orbits
When we say that a planet has a highly
eccentric orbit, we mean that:
1. it is spiraling in toward the Sun.
2. its orbit is an ellipse with the Sun at
one focus.
3. in some parts of its orbit it is much
closer to the Sun than in other
parts.
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Clicker question: Comets
Suppose a comet orbits the Sun on a highly eccentric
orbit with an average (semimajor axis) distance of 1
AU. How long does it take to complete each orbit,
and how do we know?
1. It depends on the eccentricity of the orbit, as
described by Kepler's second law.
2. 1 year, which we know from Kepler's third law.
3. Each orbit should take about 2 years, because the
eccentricity is so large.
4. It depends on the eccentricity of the orbit, as
described by Kepler's first law.
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Clicker question: Comets
Suppose a comet orbits the Sun on a highly eccentric
orbit with an average (semimajor axis) distance of 1
AU. How long does it take to complete each orbit,
and how do we know?
1. It depends on the eccentricity of the orbit, as
described by Kepler's second law.
2. 1 year, which we know from Kepler's third law.
3. Each orbit should take about 2 years, because the
eccentricity is so large.
4. It depends on the eccentricity of the orbit, as
described by Kepler's first law.
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Problem 1
• The recently discovered object Eris,
which is slightly larger than Pluto, orbits
the Sun every 560 years. What is its
average distance (or semimajor axis)
from the Sun?
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Problem 1: solution
Use Kepler’s 3rd law to find the period:
p2 = a 3
Solve for a:
a = p2/3
Take p = 560 yr
a = 67.9 AU
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Problem 2
• Halley’s comet orbits the Sun every 76
years and has an orbital eccentricity of
0.97
– Find its average distance to the Sun (i.e. its
semimajor axis)
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Problem 2: solution
Use Kepler’s 3rd law to find the period:
p2 = a 3
Solve for a:
a = p2/3
Take p = 76 yr
a = 17.9 AU
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Problem 3
Halley’s orbit is very eccentric (stretchedout ellipse), so that at perihelion it is
only about 90 million km from the Sun,
compared to more than 5 billion km at
aphelion.
– Does Halley’s comet spend most of its time
near its perihelion, aphelion, or halfway
between?
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Problem 3: Solution
Halley’s comet spends most of its time far from
the Sun near aphelion; since Kepler’s second
law says that bodies move faster when they
are closer to the Sun than when they are
farther away.
 Halley’s comet moves most slowly at
aphelion.
 Since it is moving most slowly there, it
spends more time in that part of the orbit
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Galileo Galilei (1564–1642)
The main objections of the Aristotle view
to a Sun-centered Universe were:
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•
Earth could not be moving because
objects in air (birds, clouds, ..) would
be left behind as Earth moved along
its way
•
Noncircular orbits are not “perfect”
as heavens should be
•
If Earth were really orbiting the Sun,
we would detect stellar parallax
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Overcoming the first objection
(nature of motion):
Galileo’s experiments with rolling balls and dropping
objects from a height showed that objects in air
would stay with a moving Earth.
Aristotle thought that all objects naturally come to
rest.
• Galileo showed that objects will stay in motion
unless a force acts to slow them down (Newton’s
first law of motion).
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Overcoming the second objection
(heavenly perfection)
Tycho’s observations of a comet and
a supernova already challenged
this idea.
• Using his telescope, Galileo saw:
— Sunspots on Sun
(“imperfections”)
— Mountains and valleys on the
Moon (proving it is not a perfect
sphere)
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Overcoming the third objection (parallax)
• Tycho thought he had measured stellar
distances, so lack of parallax seemed to rule out
an orbiting Earth.
• Galileo used his telescope to see that the Milky
Way is made of countless individual stars:
 showed that stars must be much farther
than Tycho thought.
 If stars were much farther away, then lack of
detectable parallax was no longer so troubling.
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The final nails in the coffin of the
geocentric model
• Two of Galileo’s earliest discoveries
contributed to the demise of the
geocentric model
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1. Galileo’s discovery of four moons orbiting Jupiter
 Galileo thus proved that
not all objects orbit Earth.
Page from Galileo’s notebook
written in 1610. His sketches
show four “stars” near Jupiter
(the circle) but in different
positions at different times (and
sometimes hidden from view).
Galileo soon realized that the
“stars” were actually moons.
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2. Galileo’s observations of phases of Venus
 proved that Venus orbits the Sun and not Earth.
In the Ptolemaic model, Venus
In reality, Venus orbits the Sun, so
orbits Earth, moving around a
from Earth we can see it in many
smaller circle on its larger orbital
different phases. This is just what
circle; the center of the smaller circle
Galileo observed, allowing him to
lies on the Earth-Sun line. If this
prove that Venus orbits the Sun.
view were correct, Venus’ phases
would range only from new to
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Astro 110-01 Lecture 8
crescent
Galileo
Galileo observed all of the following. Which
observation offered direct proof of a planet
orbiting the Sun?
1. Phases of Venus
2. The Milky Way is composed of many
individual stars
3. Four moons of Jupiter
4. Patterns of shadow and sunlight near the
dividing line between the light and dark
portions of the Moon's face
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Galileo
Galileo observed all of the following. Which
observation offered direct proof of a planet
orbiting the Sun?
1. Phases of Venus
2. The Milky Way is composed of many
individual stars
3. Four moons of Jupiter
4. Patterns of shadow and sunlight near the
dividing line between the light and dark
portions of the Moon's face
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In 1633 the Catholic
Church ordered Galileo to
recant his claim that Earth
orbits the Sun.
His book on the subject
was removed from the
Church’s index of banned
books in 1824.
Galileo was formally
vindicated by the Church
in 1992.
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What have we learned?
• How did Copernicus, Tycho, and Kepler challenge
the Earth-centered idea?
— Copernicus created a Sun-centered model;
— Tycho provided the data needed to improve this model;
— Kepler found a model that fit Tycho’s data.
• What are Kepler’s three laws of planetary motion?
1. The orbit of each planet is an ellipse with the Sun at one
focus.
2. As a planet moves around its orbit it sweeps our equal
areas in equal times.
3. More distant planets orbit the Sun at slower average
speeds:
p2 = a3.
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What have we learned?
• What was Galileo’s role in solidifying the
Copernican revolution?
— His experiments and observations overcame the
remaining objections to the Sun-centered solar
system.
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