Transcript The Milky Way - University of North Texas

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Chapter 5
Newton, Einstein, and Gravity
Guidepost
Astronomers are gravity experts. All of the heavenly motions
described in the preceding chapters are dominated by
gravitation. Isaac Newton gets the credit for discovering
gravity, but even Newton couldn’t explain what gravity was.
Einstein proposed that gravity is a curvature of space, but
that only pushes the mystery further away. “What is
curvature?” we might ask.
This chapter shows how scientists build theories to explain
and unify observations. Theories can give us entirely new
ways to understand nature, but no theory is an end in itself.
Astronomers continue to study Einstein’s theory, and they
wonder if there is an even better way to understand the
motions of the heavens.
The principles we discuss in this chapter will be companions
through the remaining chapters. Gravity is universal.
Outline
I. Galileo and Newton
A. Galileo and Motion
B. Newton and the Laws of Motion
C. Mutual Gravitation
II. Orbital Motion
A. Orbits
B. Orbital Velocity
C. Calculating Escape Velocity
D. Kepler's Laws Re-examined
E. Newton's Version of Kepler's Third Law
F. Astronomy After Newton
III. Einstein and Relativity
A. Special Relativity
B. The General Theory of Relativity
C. Confirmation of the Curvature of Space-Time
A New Era of Science
Mathematics as a tool for
understanding physics
Isaac Newton (1643 - 1727)
• Building on the results of Galileo and Kepler
• Adding physics interpretations to the
mathematical descriptions of astronomy by
Copernicus, Galileo and Kepler
Major achievements:
1. Invented Calculus as a necessary tool to solve
mathematical problems related to motion
2. Discovered the three laws of motion
3. Discovered the universal law of mutual gravitation
Velocity and Acceleration
Acceleration (a) is the change of a
body’s velocity (v) with time (t):
a
a = Dv/Dt
Velocity and acceleration are directed
quantities (vectors)!
v
Different cases of acceleration:
1. Acceleration in the conventional
sense (i.e. increasing speed)
2. Deceleration (i.e. decreasing speed)
3. Change of the direction of motion
(e.g., in circular motion)
Acceleration of Gravity
Iron ball
Wood ball
Acceleration
of gravity is
independent
of the mass
(weight) of the
falling object!
Newton’s Laws of Motion (1)
1. A body continues at
rest or in uniform
motion in a straight
line unless acted
upon by some net
force.
An astronaut floating in space
will continue to float forever in
a straight line unless some
external force is accelerating
him/her.
Newton’s Laws of Motion (2)
2. The acceleration a
of a body is
inversely
proportional to its
mass m, directly
proportional to the
net force F, and in
the same direction
as the net force.
a = F/m  F = m a
Newton’s Laws of Motion (3)
3. To every action,
there is an equal
and opposite
reaction.
M = 70 kg
V=?
The same force that is
accelerating the boy
forward, is accelerating
the skateboard backward.
m = 1 kg
v = 7 m/s
The Universal Law of Gravity
• Any two bodies are attracting each
other through gravitation, with a force
proportional to the product of their
masses and inversely proportional to
the square of their distance:
F=-G
Mm
r2
(G is the Universal constant of gravity.)
Understanding Orbital Motion
The universal law of gravity allows us to
understand orbital motion of planets and
moons:
Example:
• Earth and moon attract each other through gravitation.
• Since Earth is much more
massive than the moon, the moon’s
effect on Earth is small.
• Earth’s gravitational force
constantly accelerates the moon
towards Earth.
• This acceleration is constantly
changing the moon’s direction of
motion, holding it on its almost
circular orbit.
Dv
v
v’
Moon
F
Earth
Center of Mass
(SLIDESHOW MODE ONLY)
Orbital Motion (2)
In order to stay on a
closed orbit, an object
has to be within a
certain range of
velocities:
Too slow => Object falls
back down to Earth
Too fast => Object escapes
Earth’s gravity
Orbital Motion (3)
Geosynchronous
Orbits
Newton’s Cannon
(SLIDESHOW MODE ONLY)
Geosynchronous Orbit
(SLIDESHOW MODE ONLY)
Kepler’s Third Law Explained by
Newton
Balancing the force (called “centripetal
force”) necessary to keep an object in
circular motion with the gravitational
force  expression equivalent to Kepler’s
third law,
Py2 = aAU3
Einstein and Relativity
Einstein (1879 – 1955) noticed
that Newton’s laws of motion are
only correct in the limit of low
velocities, much less than the
speed of light.
 Theory
of Special Relativity
Also, revised understanding
of gravity
 Theory
of General Relativity
Two Postulates Leading to Special
Relativity (1)
1. Observers can
never detect their
uniform motion,
except relative to
other objects.
This is equivalent to:
The laws of physics are the same for all
observers, no matter what their motion, as
long as they are not accelerated.
Two Postulates Leading to Special
Relativity (2)
2. The velocity of
light, c, is
constant and
will be the
same for all
observers,
independent of
their motion
relative to the
light source.
Basics of Special Relativity
The two postulates of special relativity
have some amazing consequences.
Consider thought experiment:
Motion of
“stationary
” observer
v’
Assume a light source moving with velocity v
relative to a “stationary” observer:
v
c Dt’
Light
source
Seen by an observer
moving along with the light
source
v
c Dt’
c Dt
v Dt
Seen by the
“stationary” observer
Basics of Special Relativity (2)
Now, recall that the velocity of light, c, is
the same for all observers.
 The times Dt and Dt’ must be different!
Then, the Pythagorean Theorem gives:
(cDt)2 = (cDt’)2 + (vDt)2
or
c Dt
c Dt’
Dt’ = (Dt)/g
where g = 1/(1 – [v/c]2)1/2
v Dt
is the Lorentz factor.
This effect is called time dilation.
Other Effects of Special Relativity
• Length contraction: Length
scales on a rapidly moving
object appear shortened.
• Relativistic aberration:
Distortion of angles
• The energy of a body at rest
is not 0. Instead, we find
E0 = m c2
General Relativity
A new description of gravity
Postulate:
Equivalence Principle:
“Observers can not
distinguish locally
between inertial forces
due to acceleration and
uniform gravitational
forces due to the
presence of massive
bodies.”
Another Thought Experiment
Imagine a light source on board a rapidly
accelerated space ship:
Time
Light
source
Time
a
a
a
a
g
As seen by a
“stationary” observer
As seen by an observer
on board the space ship
Thought Experiment (2)
For the accelerated observer, the light
ray appears to bend downward!
Now, we can’t distinguish between
this inertial effect and the effect of
gravitational forces
Thus, a gravitational force
equivalent to the inertial force
must also be able to bend light!
Thought Experiment (Conclusion)
This bending of light by the gravitation of massive
bodies has indeed been observed:
During total solar
eclipses:
The positions of
stars apparently
close to the sun
are shifted away
from the position
of the sun.

New description of gravity as
curvature of space-time!
Another manifestation of bending of light:
Gravitational lenses
A massive galaxy cluster is bending and
focusing the light from a background object.
Other Effects of General Relativity
• Perihelion advance
(in particular, of
Mercury)
• Gravitational red shift: Light from sources near
massive bodies seems shifted towards longer
wavelengths (red).
New Terms
natural motion
violent motion
acceleration of gravity
momentum
mass
acceleration
velocity
inverse square law
field
circular velocity
geosynchronous satellite
center of mass
closed orbit
escape velocity
open orbit
angular momentum
energy
joule (J)
special relativity
general theory of relativity
Discussion Questions
1. How did Galileo idealize his inclines to conclude that
an object in motion stays in motion until it is acted on by
some force?
2. Give an example from everyday life to illustrate each
of Newton’s laws.
Quiz Questions
1. According to Aristotle, where is the proper place of the
classical elements earth and water; that is, what location do
they seek?
a. The center of Earth.
b. The center of the Universe.
c. The Heavens.
d. Both a and b above.
e. Both b and c above.
Quiz Questions
2. According to the principles of Aristotle, what part of the
motion of an arrow that is fired vertically upward is natural
motion and what part is violent motion?
a. Both the upward and downward parts are natural motion.
b. Both the upward and downward parts are violent motion.
c. The upward part is natural motion and the downward part is
violent motion.
d. The upward part is violent motion and the downward part is
natural motion.
e. Neither the upward nor the downward parts are natural or
violent motion.
Quiz Questions
3. If we drop a feather and a hammer at the same moment and
from the same height, on Earth we see the hammer strike the
ground first, whereas on the Moon both strike the ground at the
same time. Why?
a. The surface gravity of Earth is stronger than the gravity of
the Moon.
b. In strong gravity fields heavier objects fall faster.
c. The is no air resistance effect on the Moon.
d. Both a and b above.
e. All of the above.
Quiz Questions
4. Which statement below best describes the difference
between your mass and your weight?
a. Your mass is constant and your weight varies throughout
your entire life.
b. Your mass is a measure of the amount of matter that you
contain and your weight is a measure of the amount of
gravitational pull that you experience.
c. Your mass is a measure of your inertia, whereas your weight
is a measure of the amount of material you contain.
d. The only difference is the unit used to measure these two
physical quantities. Mass is measured in kilograms and weight
is measured in pounds.
e. There is no difference between your mass and your weight.
Quiz Questions
5. Which of the following is true for an object in uniform circular
motion?
a. The velocity of the object is constant.
b. The acceleration of the object is zero.
c. The acceleration of the object is toward the center of motion.
d. The angular momentum of the object is zero.
e. The speed of the object is changing.
Quiz Questions
6. If a 1-kilogram rock and a 6-kilogram rock are dropped from
the same height above the Moon's surface at the same time,
they both strike the Moon's surface at the same time. The
gravitational force with which the Moon pulls on the 6-kg rock is
6 times greater than on the 1-kg rock. Why then do the two
rocks strike the Moon's surface at the same time?
a. The acceleration of each rock is inversely proportional to its
mass.
b. The Moon's surface gravity is one-sixth the surface gravity at
Earth's surface.
c. The 1-kg rock is attracted less by the nearby Earth.
d. Both a and b above.
e. All of the above.
Quiz Questions
7. Why did Newton conclude that some force had to pull the
Moon toward Earth?
a. The Moon's orbital motion is a curved fall around Earth.
b. The Moon has an acceleration toward Earth.
c. The force and acceleration in Newton's second law must
have the same direction.
d. Both b and c above.
e. All of the above.
Quiz Questions
8. What did Newton determine is necessary for the force
exerted by the Sun on the planets to yield elliptical orbits?
a. The force must be attractive.
b. The force must be repulsive.
c. The force must vary inversely with distance.
d. The force must vary inversely with distance squared.
e. Both a and d above.
Quiz Questions
9. Which of Kepler's laws of planetary motion is a consequence
of the conservation of angular momentum?
a. The planets orbit the Sun in elliptical paths with the Sun at
one focus.
b. A planet-Sun line sweeps out equal areas in equal intervals
of time.
c. The orbital period of a planet squared is proportional to its
semimajor axis cubed.
d. Both b and c above.
e. All of the above.
Quiz Questions
10. How did Galileo slow down time in his falling body
experiments?
a. He performed the experiments near the speed of light.
b. He measured the time objects took to fall through water.
c. He used a stopwatch.
d. He rolled objects down inclines at low angles.
e. He began each fall with an upward toss.
Quiz Questions
11. Which of Newton's laws was first worked out by Galileo?
a. The law of inertia.
b. The net force on an object is equal to the product of its mass
and its acceleration.
c. The law of action and reaction.
d. The law of universal mutual gravitation.
e. Both c and d above.
Quiz Questions
12. According to Newton's laws, how does the amount of
gravitational force on Earth by the Sun compare to the amount
of gravitational force on the Sun by Earth?
a. The amount of force on Earth by the Sun is greater by the
ratio of the Sun's mass to Earth's mass.
b. The amount of force on the Sun by Earth is negligible.
c. The amount of force on the Sun by Earth is the same as the
amount of force on Earth by the Sun.
d. The amount of force on the Sun by Earth is greater by the
ratio of the Sun's mass to Earth's mass.
e. It is impossible to compare these two vastly different
amounts of force.
Quiz Questions
13. Suppose that Planet Q exists such that it is identical to
planet Earth yet orbits the Sun at a distance of 5 AU. How does
the amount of gravitational force on Planet Q by the Sun
compare to the amount of gravitational force on Earth by the
Sun?
a. The amount of the two forces is the same.
b. The amount of force on Planet Q is one-fifth the force on
Earth.
c. The amount of force on Planet Q is 5 times the force on
Earth.
d. The amount of force on Planet Q is one twenty-fifth the force
on Earth.
e. The amount of force on Planet Q is 25 times the force on
Earth.
Quiz Questions
14. Newton's form of Kepler's law can be written as: (Msun +
Mplanet) Py2 = aAU3, where the masses of the Sun and planet are
in units of solar masses, the period is in units of years, and the
semimajor axis in astronomical units. Why is Kepler's form of
his third law nearly identical to Newton's form?
a. Both forms are very similar in that they have periods and
semimajor axes in units of years and astronomical units
respectively.
b. The mass of the Sun plus the mass of a planet is nearly one.
c. The mass of each planet is very large.
d. Both b and c above.
e. All of the above.
Quiz Questions
15. How does the orbital speed of an asteroid in a circular solar
orbit with a radius of 4.0 AU compare to a circular solar orbit with a
radius of 1.0 AU?
a. The two orbital speeds are the same.
b. The circular orbital speed at 4.0 AU is four times that at 1.0 AU.
c. The circular orbital speed at 4.0 AU is twice that at 1.0 AU.
d. The circular orbital speed at 4.0 AU is one-half that at 1.0 AU.
e. The circular orbital speed at 4.0 AU is one-fourth that at 1.0 AU.
Quiz Questions
16. In the 1960s television program "Space 1999" an accident
on the Moon causes the Moon to be accelerated such that it
escapes Earth and travels into interstellar space. If you
assume that the Moon's orbit was nearly circular prior to the
accident, by what minimum factor is the Moon's orbital speed
increased?
a. The Moon's speed must be increased by a factor of 4 to
escape Earth.
b. The Moon's speed must be increased by a factor of pi to
escape Earth.
c. The Moon's speed must be increased by a factor of 2 to
escape Earth.
d. The Moon's speed must be increased by a factor of 1.4 to
escape Earth.
e. It cannot be determined from the given information.
Quiz Questions
17. Just after a alien spaceship travels past Earth at one-half
the speed of light, a person on Earth sends a beam of light past
the ship in the same direction that the ship is traveling. How
fast does an alien on the ship measure the light beam to be
traveling as it zips past the spaceship?
a. At the speed of light, or 300,000 km/s.
b. At one-half the speed of light, or 150,000 km/s.
c. At one and one-half the speed of light, or 450,000 km/s.
d. At twice the speed of light, or 600,000 km/s.
e. The measured speed depends on the method of
measurement.
Quiz Questions
18. Who first proposed that gravity is the bending of space-time
due to the presence of matter?
a. Tycho Brahe (1546 - 1601)
b. Johannes Kepler (1571 - 1630)
c. Galileo Galilei (1564 - 1642)
d. Isaac Newton (1642 - 1727)
e. Albert Einstein (1879 - 1955)
Quiz Questions
19. What major orbital problem of the late 1800s is solved by
general relativity?
a. The reason for the elliptical shape of planetary orbits.
b. The relationship between circular and escape velocity.
c. The periods of parabolic and hyperbolic orbits.
d. The excess precession of Mercury's perihelion.
e. The three-body problem.
Quiz Questions
20. What is significant about the May 29, 1919 solar eclipse?
a. It was an annular eclipse visible from South America, the
South Atlantic, and central Africa in 1919.
b. The bending of light by gravity was observed, thus verifying
general relativity.
c. The Moon was at New phase and at one node of its orbit
during this eclipse.
d. It marked the end of the first complete Saros cycle of the
20th century.
e. It was not predicted.
Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
d
d
c
b
c
a
e
e
b
d
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
a
c
d
b
d
d
a
e
d
b