Chapter 3
Courtesy of NASA, JSC Digital Image Collection
Gravity and the
Rise of Modern
Astronomy
Earth seen from the Moon
3-1 Galileo Galilei and the Telescope
1. Galileo was born in 1564 and was a contemporary of
Kepler. He built his first telescope in 1609.
2. Galileo was the first to use a telescope to study the
sky. He made five important observations that
affected the comparison between the geocentric and
heliocentric theories.
(a) Mountains and valleys on the Moon
(b) Sunspots
(c) More stars than can be observed with the naked eye
(d) Four moons of Jupiter
(e) Complete cycle of phases of Venus
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Observing the Moon, the Sun, and the Stars
1. Though Galileo’s first three observations do not
disprove the geocentric model, they cast doubt
on its basic assumption of perfection in the
heavens.
2. The existence of stars too dim to be seen with
the naked eye also cast doubt on the literal
interpretation of some Biblical passages.
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1. In 1610 Galileo discovered that Jupiter
had four satellites of its own, now known
as the Galilean moons of Jupiter.
2. The motion of Jupiter and its orbiting
moons contradicted the Ptolemaic
notions that the Earth is the center of all
things and that if the Earth moved
through space it would leave behind the
Moon.
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© Stock Montage, Inc./Alamy Images
Jupiter’s Moons
Figure 3.03c: Io and Europa in front of Jupiter
Courtesy of NASA, Voyager 2 photo/JPL
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The Phases of Venus
1. Galileo observed that Venus goes through a full
set of phases: full, gibbous, quarter, crescent.
2. Venus’s full set of phases cannot be explained by
the Ptolemaic model but can be explained by the
heliocentric model.
3. The Ptolemaic model predicts that Venus will
always appear in a crescent phase, which is not
borne out by the observations.
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Figure 3.05: Venus's motion according to Ptolemy
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4. Also, the heliocentric model explains the correlation
between Venus’ phases and its corresponding
observed sizes.
5. Galileo is credited with setting the standard for
studying nature through reliance on observation
and experimentation to test hypotheses.
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Newton’s First Two Laws of Motion
1. The year Galileo died—1642—is the year Isaac
Newton was born. Newton took the work of
Galileo and Kepler and created a new theory
of motion.
2. Newton’s First Law (Law of Inertia): Unless a
net, outside force, acts upon an object, the
object will maintain a constant speed in a
straight line (if initially moving), or remain at
rest (if initially at rest).
3. Inertia is the tendency of an object to resist a
change in its motion.
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© North Wind Picture Archives/Alamy Images
3-2 Isaac Newton’s Grand Synthesis
4. The first law indicates that a net force is
necessary for an object to change its speed
and/or its direction of motion (i.e., to accelerate).
5. Newton’s second law quantifies and extends the
first law. It tells us how much force is necessary
to produce a certain acceleration of an object.
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An Important Digression—Mass and Weight
1. Mass is the quantifiable property of an object
that is a measure of its inertia. It is an intrinsic
property of an object and independent of
location.
2. Mass is NOT volume or weight. (The weight of an
object on Earth is simply the downward force
experienced by the object due to its gravitational
interaction with the Earth.)
3. The international (SI) unit of mass is the
kilogram. A kilogram weighs about 2.2 pounds
on Earth.
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Back to Newton’s Second Law
1. Newton’s Second Law
A net external force applied to an object causes it
to accelerate at a rate that is inversely proportional
to its mass:
Acceleration = net force / mass, or F = m a.
2. When the net force is zero, there is no acceleration.
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Figure 3.07: The brick will accelerate if a
force is exerted on it. If twice as much
force is exerted on it, it will accelerate at
twice the rate.
Figure 3.08: The same amount of
force will give twice as much mass
only half the acceleration
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Newton’s Third Law
1. Newton’s Third Law: When object X exerts a force
on object Y, object Y exerts an equal and
opposite force back on X.
2. The Third Law is sometimes stated as “For every
action there is an equal and opposite reaction,”
but the first statement is more precise in terms
of physical forces.
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3-3 Motion in a Circle
1. Motion of an object in a circle at constant speed
(uniform circular motion) is an example of
acceleration causing a change in direction.
2. Centripetal (“center-seeking”) force is the force
directed toward the center of the curve along which
the object is moving. Centripetal force is simply a
label we apply to a net force that causes an object
to move in a curve.
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Figure 3.10: The string breaks as the rock is whirled in a circle. Which way
does the rock go after the string breaks?
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3-4 The Law of Universal Gravitation
1. The law of universal gravitation 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.
2. In equation form:
F = Gm1m2 / d2,
where G is a constant, m1 and m2 are the masses,
and d is the distance between their centers.
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3. Weight is the gravitational force between an
object and the planetary/stellar body where the
object is located.
4. According to Newton, gravity not only makes
objects fall to Earth but keeps the Moon in orbit
around the Earth and keeps the planets in orbit
around the Sun. His laws could explain the
planets’ motions and why Kepler’s laws worked.
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Arriving at the Law of Universal Gravitation
1. Whether or not force is proportional to mass can be
tested by showing that weight is proportional to
mass here on Earth.
2. To test the dependence of force on distance, Newton
compared accelerations of objects near the Earth’s
surface to the Moon’s acceleration in orbit around
the Earth.
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3. Because the distance
from the center of the
Earth to the Moon is
about 60 times the
distance from the center
of the Earth to its
surface, the centripetal
acceleration of the
Moon should be (1/60)2
or 1/3600 of the
acceleration of gravity
on Earth.
Newton’s calculations
showed this to be the
case and confirmed the
validity of his theory of
gravitation.
Figure 3.12: Weight decreases with distance from Earth
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3-5 Newton’s Laws and Kepler’s Laws
1. Kepler’s first law (that the planets move in
elliptical orbits) can be derived from Newton’s
laws but requires calculus.
2. Kepler’s second law (that an imaginary line
connecting a planet and the Sun sweeps 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.
3. Newton showed mathematically that Kepler’s
third law—the period-distance relationship—
derives from the inverse square law for
gravitation.
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4. Newton’s modified version of Kepler’s third law,
a3/P2 = (m1 + m2)/mSun,
is valid for any two objects orbiting each other
as a result of their mutual gravitational attraction
(a binary system). For the case where the two
objects are the Sun and a planet, Newton’s
modified version reduces to Kepler’s third law.
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3-6 Orbits and the Center of Mass
1. 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 center of
mass of a system is not always the same as the
center of gravity.
2. The center of mass for the Earth-Moon system is
inside the Earth, about 4,800 km from its center.
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Figure 3.15: Barycenter of Earth-Moon system
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3-6 Orbits and the Center of Mass
3. Historically, 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 around the center of mass.
4. When we consider the motion of the Earth
around the Sun, it is the center of mass of the
Earth-Moon system that follows an elliptical
path.
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3-7 Beyond Newton
1. Newtonian ideas succeed in describing a wide
range of observations based on a few basic ideas:
for example,
– Kepler’s laws can be derived from them,
– they explain tides and precession, and
– their use predicted the existence of the planet Neptune.
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2. Newton’s laws provide a way to measure things
quantitatively and predict the motion of things.
They were the first “laws” ever that could be shown to hold
for both the heavens and the Earth.
They offer a unifying view of the universe.
3. Newton’s work confirmed the belief of the ancient
Greeks that nature is explainable.
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3-8 General Theory of Relativity
1. We defined mass as the measure of the inertia of
an object. However, in Newton’s law of gravity,
mass determines the strength of gravitational
attraction. The same quantity measures two
seemingly different physical properties.
2. Experiments show that the measures of inertia
and gravitational attraction are identical to one
part in a trillion. This is not a coincidence.
3. Einstein, in his General Theory of Relativity,
showed mathematically that the two types of
masses are indeed equivalent.
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4. According to the principle of equivalence, effects
of acceleration are indistinguishable from
gravitational effects.
5. Space warp, or the curvature of space, is used by
Einstein to explain the similarity of gravity and
acceleration as well as the bending of light near a
massive object.
6. For Einstein, as objects move, they follow the
curvature of space created by the presence of
mass. This view differs from Newton’s attractive
gravitational force between objects.
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Figure 3.19a,b: According to the General Theory of Relativity, matter curves
the space around it.
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© AIP Emilio Segrè Visual Archives
3-9 Gravitation and Einstein
Test 1: The Gravitational Bending of
Light
1. The general theory predicts that light will
curve in the presence of a massive
object.
This prediction, made in 1915, was first
confirmed during a solar eclipse in 1919.
2. The bending of starlight by a massive
object (the Sun, for example) is fully
explained by the general theory of
relativity but is not by Newtonian
mechanics.
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Figure 3.20: Light from the two stars is bent as it passes near the Sun,
causing the stars to appear farther apart.
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Test 2: The Orbit of Mercury
1. 19th century measurements showed that Mercury’s elliptical
orbit slowly precesses about the Sun at the rate of 574
arcseconds per century.
(Precession of an orbit corresponds to the change in orientation of
the major axis of the elliptical path of an object.)
2. Calculations based on Newtonian mechanics can account for
531 arcseconds of this precession of Mercury’s orbit. However,
43 arcseconds are unaccounted for by classical theory.
3. Einstein’s general theory of relativity was able to explain the
“mysterious” 43 arcseconds of precession by considering the
properties of curved space. The Sun’s mass curves space in its
vicinity and Mercury’s orbit is affected by this space curving.
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Figure 3.21: Precession of Mercury
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Additional Tests
1. According to the general theory of relativity, massive
objects warp not only space around them but also
time
Time slows down in the presence of gravity.
2. This effect was tested in 1960 and the results
(discussed in Chapter 15) were in complete
agreement with the theory.
3. Also, the theory predicts the existence of
gravitational waves, ripples in the curvature of
space produced by changes in the distribution of
matter.
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4. Strong indirect evidence for gravitational waves
comes from observations of a binary system of stars
(discussed in Chapter 15).
5. Newton’s theory of gravity does not include any
effect that gravity might have on time and does not
predict the existence of gravitational waves.
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The Correspondence Principle
1. The correspondence principle states that the
predictions of a new theory must agree with the
theory it replaces in cases where the previous
theory has been found to be correct.
2. The general theory of relativity is in accord with
the correspondence principle.
It agrees with Newtonian mechanics where the older
theory provided correct results.
3. All tests of Einstein’s relativity theory have
confirmed it.
We still use Newtonian mechanics, however, when it fits,
because it is easier to understand it.
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Historical Note: The Special Theory of Relativity
1. The special theory of relativity is based on two
principles.
(a) All laws of physics are the same for all non-accelerating
observers, independent of their speeds.
(b) The speed of light is the same for all non-accelerating
observers, no matter what their motion relative to the
source of light.
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2. The special theory of relativity predicts that:
(a) the observed length along the line of motion of
a moving object becomes less than its length
when measured at rest;
(b) the observed passage of time becomes slower
for the moving object;
(c) the observed inertia of an object becomes
greater than its inertia when at rest; and
(d) mass can be transformed to energy and vice
versa.
3. Experimental results confirm the predictions made
by the special theory of relativity.
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Figure 3.B07: Constant speed of light
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