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Part
I: Developing Perspective
The remainder of this Instructor Guide goes through the book
chapter by chapter. Each chapter is organized as:
• A
brief introduction with general comments about the chapter.
• Teaching
Notes. Organized section by section for the chapter, these are essentially
miscellaneous notes that may be of use to you when teaching your course.
• Answers/Discussion
Points for Think About It and See It for Yourself Questions.
Chapter
1. A Modern View of the Universe
The purpose of this first chapter is to
provide students with the contextual framework they need to learn the rest of
the course material effectively: a general overview of the scale of the
universe (Section 1.1), the history of the universe and the scale of time
(Section 1.2), and an overview of motion in the universe (Section 1.3). We
often tell students that, after completing this first chapter, they have
essentially learned all the major ideas of astronomy, and the rest of their
course will be building the detailed scientific case for these general ideas.
Key Changes for the 7th Edition: For
those who have used earlier editions of our textbook, please note the following
significant changes in this chapter:
• Although
the overall content remains nearly the same as in the prior edition, we have
significantly reorganized the material to make it easier for students to follow
and understand. We have also reduced the number of Learning Goals to two in
each of the first three sections, resulting in a total of seven for the entire
chapter
(in contrast to the previous 14); this should make it much easier for students to know how to focus their study in this chapter.
(in contrast to the previous 14); this should make it much easier for students to know how to focus their study in this chapter.
• Section
1.1, which previously covered both our place in the universe and the history of
the universe, now focuses exclusively on the former. It also now includes
discussion of the scale of the universe, which previously appeared in Section
1.2.
• Section
1.2 now focuses on the history of the universe, including the scale of time.
• We
have heavily reworked the four Mathematical Insight boxes to make them shorter
and easier to follow. Note also that the Mastering Astronomy Web site now has
Math Review videos that we created to cover the same ideas, along with
additional review of topics like fractions, scientific notation, and unit
analysis.
Teaching
Notes (By Section)
Section 1.1 The Scale of the
Universe
This section provides a brief overview of
our place in the universe, including the hierarchical structure of the universe
(our cosmic address) and the scale of the universe.
• Our emphasis on scale may seem unusual to
those who have not taught from our book previously. However, we believe it is
an extraordinarily important topic that generally is underappreciated by
students. Most students enter our course without any realistic view of the true
scale of the universe, and without the context of scale, they are likely to
misinterpret much of the rest of the content and detail in
an astronomy course. That is why, in addition to the discussion of scale in this section and throughout the book, we have also included the foldout “Your Place in Space” in the front of the book, so students can refer to ideas of scale at any time.
an astronomy course. That is why, in addition to the discussion of scale in this section and throughout the book, we have also included the foldout “Your Place in Space” in the front of the book, so students can refer to ideas of scale at any time.
• Note that the section begins with a brief
discussion of the Hubble Ultra Deep Field, which helps set the context for an
overview of our place in the universe.
• Note the box on “Basic Astronomical
Definitions”: Although some of the terms in this box are not discussed
immediately, having them here in the beginning of the book should be helpful to
students. All these terms also appear in the Glossary, but they are so basic
and important that we want to emphasize them here in Chapter 1.
• This section introduces astronomical
distance measurements in AU and light-years. A couple of notes on our choices
and definitions:
• There are several different ways to define
an average distance between Earth and the Sun (e.g., averaged over phase, over
time, etc.). In defining an astronomical unit (AU) as the average
distance between Earth and the Sun,
we are using the term average to mean (perihelion + aphelion)/2, which is equivalent to the semimajor axis. This has advantages when it comes to discussing Kepler’s third law, as it is much easier for students to think of a in the equation
as average
than as semimajor axis.
we are using the term average to mean (perihelion + aphelion)/2, which is equivalent to the semimajor axis. This has advantages when it comes to discussing Kepler’s third law, as it is much easier for students to think of a in the equation
as average
than as semimajor axis.
• Note that we’ve chosen to use light-years rather than parsecs as
our primary unit for stellar and galactic distances for three reasons: (1) We
have found that light-years are more intuitive than parsecs to most students
because light-years require only an understanding of light travel times, and
not of the more complex trigonometry of parallax. (2) Lookback time is one of
the most important concepts in astronomy, and use of light-years makes it far
easier to think about lookback times (e.g., when a student hears that a star is
100 light-years away, he/she can immediately recognize that we’re seeing light
that left the star 100 years ago). (3) Fortuitously, 1 light-year happens to be
very close to 
kilometers 
making unit
conversions very easy—this helps students remember that light-years are a unit
of distance, not of time.
kilometers making unit
conversions very easy—this helps students remember that light-years are a unit
of distance, not of time.
• Note that in discussing distances to very
distant galaxies, we give distances in terms of lookback times; for example,
when we say “7 billion light-years,” we mean a lookback time of 7 billion
years. With this definition, the radius of the observable universe is about 14
billion light-years, corresponding to the age of the universe. Unfortunately,
some media reports instead quote distances as they would be today with the
ongoing expansion, hence saying that the universe is 40+ billion light-years in
radius. As a result, some students may be confused about which is the “real”
radius of the observable universe. To alleviate this confusion, we point out
that:
• “Distance” is ambiguous in an expanding
universe; e.g., do you mean an object’s distance at the time the light left, or
now, or some time in between? The choice is arbitrary, with no particular
choice (such as “now”) being any better than any other (such as “at the time
the light left”).
• In contrast, “lookback time” is
unambiguous—it is the actual amount of time that the light has been traveling
to reach us. For this reason, we feel that lookback time is a much better way
to describe distances.
• For our discussion of scale, we begin by
making use of the 1-to-10-billion scale of the Voyage scale model solar system
in Washington , D.C.
• With regard to the count to 100 billion, it
can be fun in lecture to describe what happens when you ask children how long
it would take. Young children inevitably say they can count much faster than
one per second. But what happens when they get to, say, “twenty-four billion,
six hundred ninety-seven million, five hundred sixty-two thousand, nine hundred
seventy-seven . . .”? How fast can they count now? And can they remember what
comes next?
• Regarding our claim that the number of stars
in the observable universe is roughly the same as the number of grains of sand
on all the beaches on Earth, here are the assumptions we’ve made:
• We are using 
as the number of stars
in the universe. Assuming that grains of sand typically have a volume of 
(correct within a
factor of 2 or 3), grains of sand would
occupy a volume of 
or 
as the number of stars
in the universe. Assuming that grains of sand typically have a volume of (correct within a
factor of 2 or 3), grains of sand would
occupy a volume of or
• We estimate the depth of sand on a typical
beach to be about 2–5 meters (based on beaches we’ve seen eroded by storms) and
estimate the width of a typical beach at 20–50 meters; thus, the
cross-sectional area of a typical beach is roughly 
• With
this 
cross-sectional area,
it would take a length of 
meters, or 
kilometer, to make a
volume of This is almost
certainly greater than the linear extent of sandy beaches on Earth.
cross-sectional area,
it would take a length of meters, or kilometer, to make a
volume of This is almost
certainly greater than the linear extent of sandy beaches on Earth.
Section 1.2 The History of the Universe
This section provides a brief overview of
our place in the time, including an overview of the history of the universe
(our cosmic origins) and the scale of time.
• As in the first section, we include a major
emphasis on scale—in this case, the scale of time. Be sure to point out the
backside of the foldout in the front of the book, which covers “Your Place in
Time.”
• Be sure to notice the foldout in the front
of the book; the side with “Your Place in Time” is in some sense a summary of
this section.
• We give the age of the universe as “about 14
billion years” based on the Wilkinson
Microwave Anisotropy Probe (WMAP) results
(http://map.gsfc.nasa.gov/), which are consistent with an age of 13.7 billion
years with a 1 sigma error bar of
0.2 billion years. Given this error bar, stating the age of the universe with two significant digits seems appropriate for the student audience.
0.2 billion years. Given this error bar, stating the age of the universe with two significant digits seems appropriate for the student audience.
• The idea of a “cosmic calendar” was
popularized by Carl Sagan. Now that we’ve calibrated the cosmic calendar to a
cosmic age of 14 billion years, note that
1 average month = 1.17 billion years.
1 average month = 1.17 billion years.
Section 1.3 Spaceship Earth
This section completes our overview of the
“big picture” of the universe by focusing on motion in the context of the
motions of Earth in space, using R. Buckminster Fuller’s idea of Spaceship Earth.
• Note that the summary figure for this section
comes at its end, in Figure 1.17.
• We use the term tilt
rather than obliquity as part of our continuing effort
to limit the use of jargon.
• We note that universal expansion generally
is not discussed until very late in other books. However, it’s not difficult to
understand through the raisin cake analogy; most students have heard about it
before (although few know what it means) and it’s one of the most important
aspects of the universe as we know it today. Given all that, why wait to introduce
it?
• Similarly, we briefly introduce the ideas of
dark matter and dark energy, since students have almost undoubtedly heard of
them in the media.
• With regard to our solar system’s distance
from the center of the galaxy, the literature offers values over a fairly wide
range; for example:
• 27,200 +/– 1,100 lt-yr (Gilessen et al.
2009 – from stellar orbits around central BH)
• 25,400 +/– 2,000 lt-yr (Majaess et al.
2009 – from Cepheids)
• 25,700 +/– 2,600 lt-yr (Reid et al. 2009 –
VLBI of water masers near GC)
Based on this
range, we have chosen to go with 27,000 light-years, which is the result above
with the smallest error bars and which is consistent with all three ranges.
Although
the philosophical implications of astronomical discoveries generally fall
outside the realm of science, most students enjoy talking about them. This
final section of Chapter 1 is intended to appeal to that student interest,
letting them know that philosophical considerations are important to scientists
as well.
• FYI:
Regarding the Pope’s apology to Galileo, the following is a quotation from Time magazine (Richard N. Ostling, John Moody, Rome
Popes rarely apologize. So it was big news in October when John Paul
II made a speech vindicating Galileo Galilei. In 1633 the Vatican Vatican accusers. More
than a millennium before Galileo, St.
Augustine
Answers/Discussion
Points for Think About It/See It For Yourself Questions
The Think About It and See It for Yourself
questions are not numbered in the book, so we list them in the order in which
they appear, keyed by section number.
Section 1.1
• (p. 2) This question is, of course, very
subjective, but can make for a lively in-class debate.
• (p. 4, SIFY) The main goal of this activity
is for students to go out and look for the Andromeda Galaxy and contemplate
seeing light that has been traveling through space for 2.5 million years. If
people are looking from the Andromeda Galaxy at the Milky Way, they would see a
spiral galaxy looking much like their galaxy looks to us. They would see our
galaxy as it was about 2.5 million years ago (due to light travel time) and
thus could not know that our civilization exists here today.
• (p. 10) This question can be a great topic
of debate. We’ve found that most students tend to think it is inconceivable
that we could be the only intelligent beings. However, some religious students
will assume we are alone on grounds of their faith. In both cases, it can
generate discussion about how science goes only
on evidence. For example, we don’t assume there are others because we have no evidence that there are, and we don’t assume we are alone for the same reason.
on evidence. For example, we don’t assume there are others because we have no evidence that there are, and we don’t assume we are alone for the same reason.
• (p. 10) This is another very subjective
question, but it should get students thinking about the size of Earth in the cosmos.
(It should also get them to study the scale foldout in the front of the book.)
At the least, most students are very surprised at how small our planet seems in
relation to the solar system. For most students, it makes Earth seem a little
more fragile, and often makes them think more about how we can best take care
of our planet.
Section 1.2
• (p. 14) This is another very subjective
question, similar to the one at the end of Section 1.1 but focused on our place
in time rather than in space.
Section 1.3
• (p. 15) As we authors understand it, the only
real reason that globes are oriented with north on top is because most of the
early globe makers lived in the Northern Hemisphere. In any case, it is
certainly equally correct to have the globe oriented in any other way.
• (p. 16) This question is easy to discuss if
you refer to the 1-to-10-billion scale model developed earlier. On this scale,
entire star systems are typically only a few hundred meters in diameter
(including all their planets), while they are separated from other systems by
thousands of kilometers (at least in our vicinity of the galaxy).
Solutions
to End-of-Chapter Problems (Chapter 1)
Visual Skills Check
1. b
2. c
3. c
4. No,
the nearest stars would not fit on Earth with this scale.
Review Questions
1. The
largest scale is the universe itself, which is the sum total of all matter and
energy. The largest-known organized structures are superclusters of galaxies,
then clusters and groups of galaxies, and then the roughly 100 billion
individual galaxies, most of which are many thousands of light-years across.
Each galaxy contains billions of stars, and many or most stars may be orbited
by planets.
2. Astronomical Unit: The
average distance between Earth and Sun, which is about 
kilometers.
Light-year: the distance that light travels in 1 year, which is about 9.46
trillion kilometers.
kilometers.
Light-year: the distance that light travels in 1 year, which is about 9.46
trillion kilometers.
3. Because
light travels at a fixed speed, it takes time for it to go between two points
in space. Although light travels very quickly, the distances in the universe
are so large that the time for light to travel between stars is years or
longer. The farther away we look, the longer it takes light to have traveled to
us from the objects. So the light we see from more distant objects started its
journey longer ago. This means that what we see when we look at more distant
objects is how they looked longer ago in time. So, looking farther away means
looking further back in time.
4. The
observable universe is the portion of the entire universe that we can in
principle see; it is about 14 billion light-years in radius, because light from
more than 14 billion light-years away could not yet have reached us during the
14 billion years since the Big Bang. Scientists currently think that the entire
universe is larger than the observable universe.
5. Solar
system: On the 1-to-10-billion scale, the Sun is about the size of a grapefruit
and the planets are the sizes of marbles or smaller. The distances between the
planets are a few meters for the inner solar system to many tens of meters in
the outer solar system. Distances to nearby stars: On the same scale, the
nearest stars are thousands of kilometers away. The Milky Way Galaxy: One way
to understand the size of our galaxy is to note that if the Milky Way were the
size of a football field, then the distance to the nearest star would be about
4 millimeters. The number of stars in the galaxy is more than 100 billion, so
many that it would take thousands of years to count them out loud. The
observable universe: One way to get a sense of the size of the observable
universe is to note that the number of stars in it is comparable to the number
of grains of sand on all of the beaches on the entire planet Earth.
6. When
we say that the universe is expanding, we mean that the average distance
between galaxies is increasing with time. If the universe is expanding, then if
we imagine playing time backward, we’d see the universe shrinking. Eventually,
if we went far enough back in time, the universe would be compressed until
everything were on top of everything else. This suggests that the universe may
have been very tiny and dense at some point in the distant past and has been
expanding ever since. This beginning is what we call the Big Bang. Based on
observations of the expansion rate, the Big Bang must have occurred about 14
billion years ago.
7. We
are “star stuff” because most of the atoms in our bodies (all the elements
except for hydrogen, since our bodies generally do not contain helium) were
made by stars that died long ago. These elements were released into space when
the stars died, where they were then incorporated into new generations of
stars. This is how our solar system obtained those elements, which became part
of Earth and ultimately part of us.
8. There
are numerous ways to describe how humanity fits into cosmic time, but here is
one straight from the cosmic calendar: If the entire history of the universe
were compressed into a single year, modern humans would have evolved only 2
minutes ago and the pyramids would have been built only 11 seconds ago.
9. Earth
rotates once each day on its axis and orbits the Sun once each year. The
ecliptic plane is the plane defined by Earth’s orbit around the Sun. Most of
the other planets orbit nearly in this same plane. Axis tilt is the amount that
a planet’s rotation axis is tipped relative to a line perpendicular
to the ecliptic plane.
10. The
Milky Way Galaxy is a spiral galaxy, which means that it is disk-shaped with a
large bulge in the center. The galactic disk includes a few large spiral arms.
Our solar system is located about 27,000 light-years from the center of the
galaxy, or about halfway out to the edge of the galaxy. Our solar system orbits
about the galactic center in a nearly circular orbit, making one trip around
every 230 million years.
11. The
disk of the galaxy is the flattened area where most of the stars, dust, and gas
reside. The halo is the large, spherical region that surrounds the entire disk
and contains relatively few stars and virtually no gas or dust. Dark matter
resides primarily in the halo. Dark matter and dark energy are mysterious
because even though we have strong evidence that they exist, we do not know
what they are made of.
12. Edwin
Hubble discovered that most galaxies are moving away from our galaxy, and the
farther away they are located, the faster they are moving away. While at first
this might seem to suggest that we are at the center of the universe, a little
more reflection indicates that this is not the case. If we imagine a raisin
cake rising, we can see that every raisin will move away from every other
raisin. So each raisin will see all of the others moving away from it, with
more distant ones moving faster—just as Hubble observed galaxies to be moving.
Thus, just as the raisin observations can be explained by the fact that the
raisin cake is expanding, Hubble’s galaxy observations tell us that our
universe is expanding.
Does It Make Sense?
13. Our solar system is bigger than some galaxies. This statement
does not make sense, because all galaxies are defined as collections of many (a
billion or more) star systems, so a single star system cannot be larger than a
galaxy.
14. The universe is billions of light-years in age. This
statement does not make sense, because it uses the term “light-years” as a
time, rather than as a distance.
15. It will take me light-years to complete this homework assignment.
This statement does not make sense, because it uses the term “light-years” as a
time, rather than as a distance.
16. Someday, we may build spaceships capable of traveling a light-year
in only
a decade. This statement is fine. A light-year is the distance that light can travel in 1 year, so traveling this distance in a decade would require a speed of 10% of the speed of light.
a decade. This statement is fine. A light-year is the distance that light can travel in 1 year, so traveling this distance in a decade would require a speed of 10% of the speed of light.
17. Astronomers recently discovered a moon that does not orbit a planet.
This statement does not make sense, because a moon is defined to be an object
that orbits a planet.
18. NASA soon plans to launch a spaceship that will photograph our Milky
Way Galaxy from beyond its halo. This statement does not make sense,
because of the size scales involved: Even if we could build a spaceship that
traveled close to the speed of light, it would take tens of thousands of years
to get high enough into the halo to photograph the disk, and then tens of
thousands of years more for the picture to be transmitted back to Earth.
19. The observable universe is the same size today as it was a few
billion years ago. This statement does not make sense, because the
universe is growing larger as it expands.
20. Photographs of distant galaxies show them as they were when they
were much younger than they are today. This statement makes sense,
because when we look far into space we also see far back in time. Thus, we see
distant galaxies as they were in the distant past, when they were younger than
they are today.
21. At a nearby park, I built a scale model of our solar system in which
I used a basketball to represent Earth. This statement does not make
sense. On a scale where Earth is the size of a basketball, we could not fit the
rest of the solar system in a local park. (A basketball is roughly 200 times
the diameter of Earth in the Voyage model described in the book. Since the
Earth-Sun distance is 15 meters in the Voyage model, a basketball-size Earth
would require an Earth-Sun distance of about 3 kilometers, and a Sun-Pluto
distance of about 120 kilometers.)
22. Because
nearly all galaxies are moving away from us, we must be located at the center
of the universe. This statement does not make sense, as we can tell when we
think about the raisin cake model. Every raisin sees every other raisin moving
away from it, so in this sense no raisin is any more central than any other.
(Equivalently, we could say that every raisin—or galaxy—is the center of its
own observable universe, which is true but very different from the idea of an
absolute center to the universe.)
Quick Quiz
23. a
24. b
25. c
26. b
27. a
28. b
29. c
30. b
31. a
32. a
Process of Science
33. Major
changes in scientific views are possible because science relies on physical
evidence. Science must be backed by evidence from observations or experiments,
and when the evidence does not back up the scientific story, the story is
changed. That is what happened, for example, when belief in an Earth-centered
universe gave way to the idea that Earth orbits the Sun. (In contrast, religious
or cultural beliefs generally are less subject to change, because they are
based on faith or scriptures rather than on a search for physical evidence.)
34. Using
the Voyage model scale (1 to 10 billion), Earth is barely 1 mm across but is
located 15 meters from the Sun. In other words, Earth’s distance from the Sun
is some 15,000 times as great as Earth’s diameter. Given that fact, the
difference in distance of the day and night sides of Earth is negligible, so it
would be difficult to see how it could explain day and night temperature
differences. If we had no other explanation for the temperature difference,
then we might still be forced to consider whether there is some missing piece
to our understanding. In this case, however, we have a much simpler alternative
explanation: the day side is warm because it faces the Sun, and the night side
is cooler because it faces away from the Sun.
35. Answers
will vary. This question is designed to get students thinking about the nature
of evidence and what it might take to get them to accept some scientific idea.
Group Work Exercise (no solution provided)
Short Answer/Essay Questions
37. This
is a short essay question. Key points should include discussion of the
difference in scale between interstellar travel and travel about our own world,
so that students recognize that alien technology would have to be far more
advanced than our own to allow them to visit us with ease.
38. a. The diagrams should be much like Figure
1.16, except that the distances between raisins in the expanded figure will be
4 centimeters instead of
3 centimeters.
3 centimeters.
|
b.
|
Distances and Speeds of Other Raisins
as Seen from the Local Raisin
|
|||
|
|
Raisin Number
|
Distance Before Baking
|
Distance After Baking
(1 hour later) |
Speed
|
|
|
1
|
1 cm
|
4 cm
|
3 cm/hr
|
|
|
2
|
2 cm
|
8 cm
|
6 cm/hr
|
|
|
3
|
3 cm
|
12 cm
|
9 cm/hr
|
|
|
4
|
4 cm
|
16 cm
|
12 cm/hr
|
|
|
 |
|
|
|
|
|
10
|
10 cm
|
40 cm
|
30 cm/hr
|
|
|
|
|
|
|
c. As
viewed from any location inside the cake, more distant raisins appear to move
away at faster speeds. This is much like what we see in our universe, where
more distant galaxies appear to be moving away from us at higher speeds. Thus,
we conclude that our universe, like the raisin cake, is expanding.
39. a. This problem asks students to draw a
sketch. Using the scale of 1 centimeter = 100,000 light-years, the sketches
should show that each of the two galaxies is about 1 centimeter in diameter and
that the Milky Way and M31 are separated by about 25 centimeters.
b. The separation between the Milky Way and M31
is only about 25 times their respective diameters—and other galaxies in the
Local Group lie in between. In contrast, the model solar system shows that, on
a scale where stars are roughly the size of grapefruits, a typical separation
is thousands of kilometers (at least in our region of the galaxy). Thus, while
galaxies can collide relatively easily, it is highly unlikely that two
individual stars will collide. Note: Stellar collisions
are more likely in places where stars are much closer together, such as in the
galactic center or in the centers of globular clusters.
40. This
is a subjective essay question. Grade should be based on clarity of the essay.
Quantitative Problems
41. a. A light-second is the distance that light
travels in 1 second. We know that light travels at a speed of 300,000 kilometers per second,
so a light-second is a distance
of 300,000 kilometers.
b. A light-minute is the speed of light
multiplied by 1 minute:
That
is, “1 light-minute” is just another way of saying “18 million kilometers.”
c. Following a similar procedure, we find that
1 light-hour is 1.08 billion kilometers; and
d. 1 light-day is 
kilometers, or about
26 billion kilometers.
kilometers, or about
26 billion kilometers.
42. Recall
that
We can rearrange
this with only a little algebra to solve for time:
The speed of light
is 
kilometers per second.
(We choose the value in kilometers per second rather than meters per second
because we are given distances in kilometers.)
kilometers per second.
(We choose the value in kilometers per second rather than meters per second
because we are given distances in kilometers.)
a. Mars at its nearest is 56 million kilometers
away, so the light travel time is:
We would prefer this answer in minutes, so
converting:
It takes a little over 3 minutes each way to
communicate with a spacecraft on Mars at closest approach.
b. At the most distant, Mars is 400 million
kilometers from Earth, so we can compute the travel time for light:
We would again prefer this in minutes, so we
convert:
It takes a little over 22 minutes each way to
communicate with a spacecraft on Mars when Mars is at its farthest from Earth.
c. We will again use the speed-time-distance
relationship:
In this case, we seek the time it takes light to travel from Earth
to Pluto. Earth’s average distance from the Sun is kilometers and Pluto’s
average distance is 
kilometers. If we
assume the two problems are lined up on the same side of the Sun at their
average distances, the distance between them is:
kilometers and Pluto’s
average distance is kilometers. If we
assume the two problems are lined up on the same side of the Sun at their
average distances, the distance between them is:
Using this value, the light travel time is:
This is nearly 20,000 seconds, so let’s try
changing to more comprehensible units. We’ll start by converting to minutes:
That’s a lot better, but it’s still a lot more
than 1 hour, so let’s convert to hours:
It would take light 5.33 hours, or 5 hours and
20 minutes, to travel from Earth to Pluto under the alignment conditions and
average distances we’ve assumed.
43. We are asked to find
how many times larger the Milky Way Galaxy is than the planet Saturn’s rings.
We are told that Saturn’s rings are about 270,000 kilometers across and that
the Milky Way is 100,000 light-years. Clearly, we’ll have to convert one set of
units or the other. Let’s change light-years for kilometers. In Appendix A we
find that 1 
kilometers, so we can
convert:
kilometers, so we can
convert:
We can now find the
ratio of the two diameters:
The diameter of the
Milky Way Galaxy is about 3.5 trillion times as large as the diameter of
Saturn’s rings!
44. We are given the
distance to Alpha Centauri as 4.4 light-years, but those units aren’t very
useful to us. So let’s convert to kilometers. From Appendix A we see that 1 kilometers. So the
distance to Alpha Centauri is
kilometers. So the
distance to Alpha Centauri is
At a scale of 1 to
the distance to Alpha
Centauri on this scale is
the distance to Alpha
Centauri on this scale is
As numbers go, that one’s not very easy to picture.
So let’s convert it to some smaller units. Since there are a thousand 
meters to a kilometer
and a thousand millimeters to a
meter, there are 
millimeters to a
kilometer. Those units seem about right for this distance, so let’s convert:
meters to a kilometer
and a thousand millimeters to a
meter, there are millimeters to a
kilometer. Those units seem about right for this distance, so let’s convert:
From Appendix E, the Sun is 
kilometers in radius,
so the diameter is twice this, or 
kilometers. On the 
scale, this is
kilometers in radius,
so the diameter is twice this, or kilometers. On the scale, this is
We’re asked to compare this to the size of an atom, 
meter, so we had
better convert one of the numbers. We’ll convert from kilometers to meters:
meter, so we had
better convert one of the numbers. We’ll convert from kilometers to meters:
Note that this is
about the same size as a typical atom. In summary, we’ve found that on a scale
on which the Milky Way Galaxy would fit on a football field, the distance from
the Sun to Alpha Centauri is only about 4.2 millimeters—smaller than the width
of a finger—and the Sun itself becomes as small as an atom.
45. a. First, we need to work out the conversion
factor to use to go between the real universe and our model. We are told that
the Milky Way Galaxy is to be about 1 centimeter in the new model. We know from
this chapter that the Milky Way Galaxy is about 
light-years across. We
could convert this to kilometers, but looking at what we’re asked to convert,
it appears that all the numbers we’ll be using will be given in light-years
anyway. So our conversion factor is:
light-years across. We
could convert this to kilometers, but looking at what we’re asked to convert,
it appears that all the numbers we’ll be using will be given in light-years
anyway. So our conversion factor is:
So for the first part, we need to figure out
how far the Andromeda Galaxy (also known as M31) is from the Milky Way. The
chapter tells us that their actual separation is 2.5 million light-years. So
the distance on our scale is:
The Andromeda Galaxy would be 25 centimeters
away from our galaxy on this scale.
b. From Appendix E, we learn that Alpha
Centauri is 4.4 light-years from the Sun. So using the conversion factor we
developed in part (a), we convert this to the scale model distance:
The separation between Alpha Centauri and our
Sun is about 
centimeter on this
scale. As we will see in Chapter 6, this is comparable to the wavelength of
blue light.
centimeter on this
scale. As we will see in Chapter 6, this is comparable to the wavelength of
blue light.
c. The observable universe is about 14 billion
light-years in radius, so let’s use that distance as an approximation to the
distance of the most distant observable galaxies. Converting this distance to
our model’s scale:
That’s more than a million centimeters, but
it’s difficult to visualize how far that is offhand. Let’s convert that number
to kilometers. We’ll convert to meters first since we know how to go from
centimeters to meters and meters
to kilometers, but it’s difficult to remember how to go straight from centimeters to kilometers:
to kilometers, but it’s difficult to remember how to go straight from centimeters to kilometers:
On a scale where our entire Milky Way Galaxy
is the size of a marble, the most distant galaxies in our observable universe
would be located some
14 kilometers away.
14 kilometers away.
46. a. The circumference of Earth is 
At a speed of
100 kilometers per hour, it would take:
At a speed of 100 kilometers per hour, it would take:
to drive around Earth. That is, a trip around
the equator at 100 kilometers per hour would take a little under 17 days.
b. We find the time by dividing the distance
to the planet from the Sun by the speed of 100 kilometers per hour. It would
take about 170 years to reach Earth and about 6700 years to reach Pluto (at
their mean distances).
c. Similarly, it would take 6700 years to
drive to Pluto at 100 kilometers per hour. FYI: The following table shows the
driving times from the Sun to each of the planets at a speed of 100 kilometers
per hour.
|
Planet
|
Driving Time
|
|
Mercury
|
66 yr
|
|
Venus
|
123 yr
|
|
Earth
|
170 yr
|
|
Mars
|
259 yr
|
|
Jupiter
|
888 yr
|
|
Saturn
|
1630 yr
|
|
Uranus
|
3300 yr
|
|
|
5100 yr
|
|
Pluto
|
6700 yr
|
d. We are given the distance to Alpha Centauri
in light-years; converting to kilometers, we get:
At a speed of 100 kilometers per hour, the
travel time to Proxima Centauri would be about:
It would take some 47 million years to reach
Proxima Centauri at a speed of 100 kilometers per hour.
47. a. To reach Alpha Centauri in 100 years, you
would have to travel at 
of the speed of light,
which is about 13,200 kilometers per second or nearly 50 million kilometers per
hour.
of the speed of light,
which is about 13,200 kilometers per second or nearly 50 million kilometers per
hour.
b. This is about 1000 times the speed of our
fastest current spacecraft.
48. The
average speed of our solar system in its orbit of the Milky Way is the
circumference of its orbit divided by the time it takes for one orbit:
We are racing
around the Milky Way Galaxy at nearly 800,000 kilometers per hour.
49. We’ll
use the relationship that says that
To compute each speed, we’ll find the distance a
person travels around Earth’s axis in 1 day (24 hours). Using the hint from the
problem, we can find the radius of the circle that a person at that latitude
travels is 
where 
from Appendix E.
where from Appendix E.
a. The radius of the circle that a person at
latitude 30° travels is:
To get the distance
traveled, we use the fact that a circle’s circumference is given by 
so in this case:
so in this case:
Using the relationship between speed, time,
and distance given above, the speed is
A person at 30° latitude would be traveling at 1446 kilometers per hour around
Earth’s axis because of Earth’s rotation.
b. The radius of the circle that a person at
60° travels is
The distance traveled is
The speed is
A person at 60° latitude travels around the
axis at 834 kilometers per hour due to Earth’s rotation.
c. Answers will vary depending on location.
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