Transcript DTU 8e Lecture PPT Chap 2 Gravitation v2

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?
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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.
The retrograde motion of Mars as it
would be seen in a series of images
taken on the same photographic plate.
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 January 25, 2012, through April 13, 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.
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) and Johannes Kepler
(1571–1630)
Tycho (depicted within the portrait of Kepler) 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. 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.
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, which appeared in 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 orbits around
the Sun as well as the planet.
The apparent
change in the
location of an
object due to the
difference in
location of the
observer is called
parallax.
When a new “star” appeared in the sky during the 16th century, 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 fixed and unchangeable.
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.
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.
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°.
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.
A Demonstration of Kepler’s Third Law
A Parsec
The parsec, a unit of length commonly used by
astronomers, is equal to 3.26 ly. 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.
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. Note that the
phases correlate with the planet’s angular size and its angular
distance from the Sun, both 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 alongside an
overexposed image of 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.
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.
Conservation of Angular Momentum
As this skater brings her arms and outstretched leg in,
she must spin faster to conserve her angular momentum.
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. If the object can spin around a fixed axis, like a globe,
then the rotation axis is the rod running through it. If the object is not held in
place, then the rotation axis is in a line through a point called the object’s
center of mass. The center of mass of any object is the point that follows a
smooth, elliptical path as the object moves in response to a gravitational field.
All other points in the spinning object wobble as it moves.
NEWTON’S LAW OF GRAVITATION
F
Gm1m2
r2
Gm1m2
F
2
r
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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
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. During the twentieth century, the comet passed near the Sun
twice—once in 1910 and again, as shown here, in 1986. The comet
will pass close to the Sun again in 2061. Although dim in 1986, it
nevertheless spread more than 5° across the sky, or 10 times the
diameter of the Moon.
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.
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
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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
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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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