Neil F. Comins • William J. Kaufmann III
Discovering the Universe
Ninth Edition
CHAPTER 2
Gravitation and the
Motion of the Planets
WHAT DO YOU THINK?
1.
2.
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4.
5.
6.
7.
What makes a theory scientific?
What is the shape of Earth’s orbit around the Sun?
Do the planets orbit the Sun at constant speeds?
Do all of the planets orbit the Sun at the same
speed?
How much force does it take to keep an object
moving in a straight line at a constant speed?
How does an object’s mass differ when measured
on Earth and on the Moon?
Do astronauts orbiting Earth feel the force of gravity
from our planet?
In this chapter you will discover…
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what makes a theory scientific
the scientific discoveries that revealed that Earth is not at
the center of the universe, as previously believed
Copernicus’s argument that the planets orbit the Sun
why the direction of motion of each planet on the
celestial sphere sometimes appears to change
that Kepler’s determination of the shapes and other
properties 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 scientific method is used to develop new scientific theories.
Scientific theories are accepted when they make testable
predictions that can be verified using new observations and
experiments. A theory can NEVER be proven – only disproven.
General Motion of the Planets
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Planets, to the eyeball, look virtually the same as stars.
Planets rise in the East and set in the West, same as
stars.
Over the course of a night, the relative motion of the
planet on the celestial sphere is hardly noticeable.
But, over days, the planets DO shift their position relative
to the fixed stars.
Planets move through the zodiac. They don’t follow the
ecliptic precisely, but never stray far from it.
Their motion is complicated; they sometimes move in
different directions around the zodiac.
The retrograde motion of Mars as it would be seen in a series of
images taken on the same photographic plate every 3 days.
To help visualize this motion on the celestial sphere, astronomers often plot the
position of Mars (or another body in retrograde motion) on a star chart. From
Jan 25 through Apri13 of 2012, Mars undergoes retrograde motion as seen
from Earth. The retrograde path is sometimes a loop north (shown here) or
south of the normal path, and sometimes an S-shaped path above or below it.
Retrograde Motion –Geometrical Explanation c. 150 AD
Each planet revolves around an epicycle, which in turn revolves
around a deferent centered approximately on Earth. As seen
from Earth, the speed of the planet on the epicycle alternately
(a) adds to or (b) subtracts from the speed of the epicycle on
the deferent, thus producing alternating periods of direct and
retrograde motions.
Nicolaus Copernicus developed the first complete heliocentric
(Sun-centered) model of the solar system. In this model, the
retrograde motion of Mars is seen when Earth passes Mars in its
orbit around the Sun.
Nicolaus Copernicus (1473–1543)
Copernicus, the youngest of four
children, was born in Torun, Poland.
He pursued his higher education in
Italy, where he received a doctorate in
canon law and studied medicine.
Copernicus developed a heliocentric
theory of the known universe and just
before his death in 1543 published this
work under the title De Revolutionibus
Orbium Coelestium. His revolutionary
theory was flawed in that he assumed
that the planets had circular orbits
around the Sun. This was corrected by
Johannes Kepler.
Tycho Brahe (1546–1601)
Tycho was born to nobility in the Danish city of
Knudstrup, which is now part of Sweden. At
age 20, he lost part of his nose in a duel and
wore a metal replacement thereafter. In 1576,
the Danish king Frederick II built Tycho an
astronomical observatory that Tycho
named Uraniborg (after Urania, Greek muse
of astronomy). Tycho rejected both
Copernicus’s heliocentric theory and the
Ptolemaic geocentric system. He devised a
halfway theory called the Tychonic system.
According to Tycho’s theory, Earth is
stationary, with the Sun and Moon revolving
around it, while all the other planets revolve
around the Sun. Tycho died in 1601.
Johannes Kepler (1571–1630)
.Kepler was educated in Germany,
where he spent 3 years studying
mathematics, philosophy, and
theology. In 1596, Kepler published
a booklet in which he attempted to
mathematically predict the
planetary orbits. Although his
theory was altogether wrong, its
boldness and originality attracted
the attention of Tycho Brahe,
whose staff Kepler joined in 1600.
Kepler deduced his three laws from
Tycho’s observations.
Inset portrait: Tycho Brahe
Galileo Galilei (1564–1642)
Born in Pisa, Italy, Galileo studied medicine
and philosophy at the University of Pisa. He
abandoned medicine in favor of mathematics.
He held the chair of mathematics at the
University of Padua, and eventually returned to
the University of Pisa as a professor of
mathematics. There Galileo formulated his
famous law of falling bodies: All objects fall
with the same acceleration regardless of their
weight. In 1609 he constructed a telescope
and made a host of discoveries that
contradicted the teachings of Aristotle and the
Roman Catholic Church. He summed up his
life’s work on motion, acceleration, and gravity
in the book Dialogues Concerning the Two
Chief World Systems, published in 1632.
Isaac Newton (1642–1727)
Newton delighted in constructing mechanical devices, such as sundials, model
windmills, a water clock, and a mechanical
carriage. He received a bachelor’s degree
in 1665 from the University of Cambridge.
While there, he began developing the mathematics that later became calculus (developed independently by the German Gottfried
Leibniz). While pursuing experiments in
optics, Newton constructed a reflecting
telescope and also discovered that white light is actually a
mixture of all colors. His major work on forces and gravitation was
the tome Philosophiae Naturalis Principia Mathematica (1687). In
1704, Newton published his second great treatise, Opticks, in
which he described his experiments and theories about light and
color. Upon his death in 1727, Newton was buried in Westminster
Abbey, the first scientist to be so honored.
We define special positions of the planets in their orbits
depending upon where they appear in our sky. For
example, while at a conjunction, a planet will appear in the
same part of the sky as the Sun, while at opposition, a
planet will appear opposite the Sun in our sky.
However, the cycle of these positions (a synodic period) is
different from the actual orbital period of the planet around the
Sun (a sidereal period) because Earth also orbits the Sun.
Visualize Earth on the inside for the synodic period of Mars, etc.
Mercury has an especially eccentric orbit around the Sun. As
seen from Earth, the angle of Mercury at greatest elongation
ranges from 18° to 28°. In contrast, Venus’s orbit is nearly
circular, with both greatest elongations of 47°.
Parallax
The apparent change in
the location of an object
due to the difference in
location of the observer
is called parallax.
If … the
supernova
were nearby.
Actual
Observation:
Supernova is in
the Heavens
When a new “star” appeared in the sky in 1572, Tycho Brahe
reasoned that the distance of the object may be determined by
direct measurements by examining the amount of parallax.
Because the parallax of the “star” was too small to measure,
Tycho knew that it had to be among the other stars, thus
disproving the ancient belief that the “heavens” were unchanging.
A Parsec
The parsec, a unit of length commonly used
by astronomers, is equal to 3.26 light-years.
The parsec is defined as the distance at
which 1 AU perpendicular to the observer’s
line of sight makes an angle of 1 arcsec.
One parsec = 308,000 AU
An ellipse can be drawn with a pencil, a loop of string, and two
thumbtacks, as shown. If the string is kept taut, the pencil
traces out an ellipse. The two thumbtacks are located at the
two foci of the ellipse.
Eccentricity – amount of deviation from circular
The amount of elongation in a planet’s orbit is defined as
its orbital eccentricity. An orbital eccentricity of 0 is a
perfect circle, while an eccentricity close to 1.0 is nearly a
straight line. In an elliptical orbit, the distance from a planet
to the Sun varies. The point in a planet’s orbit closest to
the Sun is called perihelion and the point in a planet’s orbit
farthest from the Sun is called aphelion.
Kepler’s first law: The orbit of a planet about the Sun is an
ellipse with the Sun at one focus.
Kepler’s second law: A line joining the planet and the Sun
sweeps out equal areas in equal intervals of time.
Kepler’s Third Law
The square of the orbital period equals
the cube of the semimajor axis
P2 = a3
A Demonstration
First observed
By Galileo
The appearance (phase) of Venus changes as it
moves along its orbit. The number below each view is
the angular diameter (d) of the planet as seen from
Earth, in arcseconds. The phases correlate with the
planet’s angular size and its angular distance from the
Sun, as seen from Earth. These observations clearly
support the idea that Venus orbits the Sun.
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. The
observations shown
were made by Jesuits in
1620 of Jupiter and its
four visible moons.
This is a photograph of the four Galilean satellites with
an overexposed Jupiter. Each satellite would be bright
enough to be seen with the unaided eye were it not
overwhelmed by the glare of Jupiter.
NEWTON’S THREE LAWS OF MOTION
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LAW #1: A body remains at rest or moves in a
straight line at constant speed unless acted
upon by a net outside force.
LAW #2: The acceleration of an object is
proportional to the force acting on it and is
inversely proportional to its mass.
FNET = m a or a = FNET/m
LAW #3: Whenever one body exerts a force on a
second body, the second body exerts an equal
and opposite force on the first body.
What is “Acceleration”
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Rate of increase of speed
Rate of decrease of speed
Rate of turning
In general, how fast velocity vector is changing
Special case: for an object moving in a circle at
constant speed:
a = v2/R, toward center of circle
Angular Momentum and Torque
(a) When a force acts through an object’s rotation axis
or toward its center of
mass, the force does not exert a torque on the object.
(b) When a force acts in some other direction, then it
exerts a torque, causing the body’s angular momentum
to change.
Conservation of Angular Momentum
As this skater brings her arms and outstretched leg in,
she must spin faster to conserve angular momentum.
NEWTON’S LAW OF GRAVITATION
Gm1m2
r2
Gm1m2
F
2
r
F
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G is a constant showing the strength of gravity; m1
and m2 are masses; and r is the distance between
the centers of the objects
Using the law of gravitation, Newton was able to derive
Kepler’s laws of motion
Watching the apple fall made Newton think “why?”
Why Does a Satellite Orbit?
1. Imagine dropping a rock. It falls 5 m is 1 second.
2. Imagine throwing the rock horizontally at fastball
speed (40 m/s) from a position on a tower. The
rock still drops 5 m in 1 second, but it moves 40 m
horizontally.
3. Now, crank the speed up to 8000 m/s. In one
second the rock drops 5 m and moves 8000 m
horizontally. The difference is that Earth’s curvature
is not negligible. In 8 km, the Earth curves by 5 m, so
the rock is at its original distance above the surface.
4. Presto! An orbit.
5. Of course, it needs to be above the atmosphere so
that the horizontal speed stays at 8 km/s.
Calculation of a Circular Orbit
The only force of consequence is Earth’s gravitation,
so FNET = FGRAV. Equate Newton’s 2nd Law with FGRAV
msat asat = G mEarth msat / r2
where “sat” refers to the satellite. The r is the distance
from the CENTER of the Earth to the satellite.
In circular motion, the acceleration is v2/r. Substitute:
msat v2/r = G mEarth msat / r2
v2 = G mEarth / r
or
Simplify:
v = sqrt(G mEarth / r)
For an orbit around the Sun, replace mEarth by mSun
The satellite (or Space Shuttle) is NOT free of gravity.
Weightlessness
So, how do astronauts float around if they aren’t free
of gravity??
They, along with the shuttle, are orbiting.
You sense weight because of two factors:
a. Earth’s gravity pulls on you
b. The floor (or chair, etc) counters its pull.
Factor (b) is missing for the astronauts. It’s like being
in an elevator and the cable breaks. You would have a
sense of weightlessness as you fall.
(By the way, as a technological note, an elevator
doesn’t actually fall if its cable breaks – thank you,
Elisha Graves Otis.)
Conic Sections
A conic section is any one of a family of curves
obtained by slicing a cone with a plane, as shown. The
orbit of one body around another can be an ellipse, a
parabola, or a hyperbola. Circular orbits are possible
because a circle is just an ellipse for which both foci
are at the same point.
Halley’s Comet
Halley’s Comet orbits the
Sun with an average period
of about 76 years. Halley
recognized its periodic
nature in 1705.
Neptune was discovered
in 1845 by Adams and
Leverrier independently
by calculations on Uranus’
orbit. It wasn’t seen until
1846.
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.
Gravity is universal.
PLANET ORBITS ARE ONLY
SLIGHTLY ELLIPTICAL
White Dashed lines:
circles centered on
the Sun
Red: Mars’ orbit
Blue: Earth’s orbit
Summary of Key Ideas
Science: Key to Comprehending the Cosmos
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The ancient Greeks laid the groundwork for
progress in science by stating that the universe is
comprehensible.
The scientific method is a procedure for formulating
theories that correctly predict how the universe
behaves.
A scientific theory must be testable, that is, capable
of being disproved.
Theories are tested and verified by observation or
experimentation and result in a process that often
leads to their refinement or replacement and to the
progress of science.
Observations of the cosmos have led astronomers
to discover some fundamental physical laws of the
universe.
Origins of a Sun-centered Universe
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Common sense (for example, Earth doesn’t appear
to be moving) led early natural philosophers to
devise a geocentric cosmology, which placed Earth
at the center of the universe.
Copernicus’s heliocentric (Sun-centered) theory
simplified the general explanation of planetary
motions compared to the geocentric theory.
The heliocentric cosmology refers to motion of
planets and smaller debris orbiting the Sun. Other
stars do not orbit the Sun.
The sidereal orbital period of a planet is measured
with respect to the stars, and determines the length
of the planet’s year. A planet’s synodic period is
measured with respect to the Sun as seen from the
moving Earth (for example, from one opposition to
the next).
Kepler’s and Newton’s Laws - I
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Ellipses describe the paths of the planets around
the Sun much more accurately than do the circles
used in previous theories. Kepler’s three laws give
important details about elliptical orbits.
The invention of the telescope led Galileo to new
discoveries, such as the phases of Venus and the
moons of Jupiter, that supported a heliocentric view
of the universe.
Newton based his explanation of the universe on
three assumptions, now called Newton’s laws of
motion. These laws and his law of universal
gravitation can be used to deduce Kepler’s laws
and to describe most planetary motions with
extreme accuracy.
Kepler’s and Newton’s Laws - II
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The mass of an object is a measure of the
amount of matter in it; weight is a measure of the
force with which the gravity of a world pulls on
an object’s mass when the two objects are at
rest with respect to each other (or, equivalently,
how much the object pushes down on a scale).
The path of one astronomical object around
another, such as that of a comet around the
Sun, is an ellipse, a parabola, or a hyperbola.
Ellipses are bound orbits, while objects with
parabolic and hyperbolic orbits fly away, never
to return.
Key Terms
acceleration
angular momentum
aphelion
astronomical unit
configuration
conjunction
conservation of angular
momentum
conservation of linear
momentum
cosmology
direct motion
ellipse
elongation
focus (of an ellipse)
force
Galilean moons
gravity
heliocentric cosmology
hyperbola
inferior conjunction
Kepler’s laws
kinetic energy
law of equal areas
law of inertia
law of universal
gravitation
light-year
mass
model
moment of inertia
momentum
Newton’s laws of
motion
Occam’s razor
opposition
parabola
parallax
parsec
perihelion
potential energy
retrograde motion
scientific method
scientific theory
semimajor axis
sidereal period
superior conjunction
synodic period
theory
universal constant of
gravitation
velocity
weight
work
WHAT DID YOU THINK?
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What makes a theory scientific?
A theory is an idea or set of ideas
proposed to explain something about the
natural world. A theory is scientific if it
makes predictions that can be objectively
tested and potentially disproved.
WHAT DID YOU THINK?
What is the shape of Earth’s orbit around
the Sun?
 All planets have elliptical orbits around the
Sun.
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WHAT DID YOU THINK?
Do the planets orbit the Sun at constant
speeds?
 No. The closer a planet is to the Sun in its
elliptical orbit, the faster it is moving. The
planet moves fastest at perihelion and
slowest at aphelion.
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WHAT DID YOU THINK?
Do all of the planets orbit the Sun at the
same speed?
 No. A planet’s speed depends on its
average distance from the Sun. The
closest planet moves fastest; the most
distant planet moves slowest.
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WHAT DID YOU THINK?
How much force does it take to keep an
object moving in a straight line at a
constant speed?
 Unless an object is subject to an outside
force, like friction, it takes no force at all to
keep it moving in a straight line at a
constant speed.
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WHAT DID YOU THINK?
How does an object’s mass differ when
measured on Earth and on the Moon?
 Assuming the object doesn’t shed or
collect pieces, its mass remains constant
whether on Earth or on the Moon. Its
weight, however, is less on the Moon.
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WHAT DID YOU THINK?
Do astronauts orbiting Earth feel the force
of gravity from our planet?
 Yes. They are continually pulled
Earthward by gravity, but they continually
miss it because of their motion around it.
Because they are continually in free fall,
they feel weightless.
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