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Chapter 2. Discovering the Universe for Yourself
This chapter introduces major phenomena of
the sky, with emphasis on:
• The concept of the celestial sphere.
• The basic daily motion of the sky, and how
it varies with latitude.
• The cause of seasons.
• Phases of the Moon and eclipses.
• The apparent retrograde motion of the
planets, and how it posed a problem for ancient observers.
As
always, when you prepare to teach this chapter, be sure you are familiar with
the online quizzes, interactive figures and tutorials, assignable homework, and
other resources available on the MasteringAstronomy Web site.
• We
have edited throughout the chapter to improve clarity for students, including
changes to several of the annotated figures.
• We have reworked the introduction to Section 2.3 and Figure 2.21
to focus more clearly on the scale of the Moon’s orbit and why sunlight
reaching it is essentially coming in with parallel rays.
• We have added two new See It for Yourself activities, one each in
Sections 2.3 and 2.4, designed to encourage students to make naked eye sky
observations.
• We
have updated the discussion of eclipses, including revising the Table 2.1 and
Figure 2.30 of upcoming eclipses.
Teaching
Notes (By Section)
Section 2.1 Patterns in the Night
Sky
This section introduces the concepts of
constellations and of the celestial sphere, and introduces horizon-based
coordinates and daily and annual sky motions.
• Stars in the daytime: You may be surprised at
how many of your students actually believe that stars disappear in the daytime.
If you have a campus observatory or can set up a small telescope, it’s well
worth offering a daytime opportunity to point the telescope at some bright
stars, showing the students that they are still there.
• In class, you may wish to go further in
explaining the correspondence between the Milky Way Galaxy and the Milky Way in
our night sky. Tell your students to imagine being a tiny grain of flour inside
a very thin pancake (or crepe!) that bulges in the middle and a little more
than halfway toward the outer edge. Ask, “What will you see if you look toward
the middle?” The answer should be “dough.” Then ask what they will see if they
look toward the far edge, and they’ll give the same answer. Proceeding
similarly, they should soon realize that they’ll see a band of dough encircling
their location, but that if they look away from the plane, the pancake is thin
enough that they can see to the distant universe.
• Sky variation with latitude: Here, the
intention is only to give students an overview of the idea and the most basic
rules (e.g., latitude = altitude of NCP or SCP). Those instructors who want
their students to be able to describe the sky in detail should cover Chapter
S1, which covers this same material, but in much more depth.
• Note that in our jargon-reduction efforts,
we do not introduce the term asterism, instead speaking
of patterns of stars in the constellations. We also avoid the term azimuth when discussing horizon-based coordinates. Instead,
we simply refer to direction along the horizon (e.g.,
south, northwest). The distinction of “along the horizon” should remove
potential ambiguity with direction on the celestial sphere (where “north” would
mean toward the north celestial pole rather than toward the horizon).
Section 2.2 The Reason for Seasons
This section focuses on seasons and why
they occur.
• In combating misconceptions about the cause
of the seasons, we recommend that you follow the logic in the Common
Misconceptions box. That is, begin by asking your students what they think
causes the seasons. When many of them suggest it is linked to distance from the
Sun, ask how seasons differ between the two hemispheres. They should then see
for themselves that it can’t be distance from the Sun, or seasons would be the
same globally rather than opposite in the two hemispheres.
• As a follow-up on the above note: Some
students get confused by the fact that season diagrams (such as our Figure
2.15) cannot show the Sun-Earth distance and size of Earth to scale. Thus,
unless you emphasize this point (as we do in the figure), it might actually
look like the two hemispheres are at significantly different distances from the
Sun. This is another reason why we believe it is critical to emphasize ideas of
scale throughout your course. In this case, use the scale model solar system as
introduced in Section 1.2, and students will quickly see that the two
hemispheres are effectively at the same distance from the Sun at
all times.
• Note that we do not go deeply
into the physics that causes precession, as even a basic treatment of this
topic requires discussing the vector nature of angular momentum. Instead, we
include a brief motivation for the cause of precession by analogy to a spinning
top.
• FYI regarding Sun signs: Most astrologers
have “delinked” the constellations from the Sun signs. Thus, most astrologers would
say that the vernal equinox still is in Aries—it’s just that Aries is no longer
associated with the same pattern of stars as it was in a.d. 150. For a fuller treatment of issues associated with
the scientific validity (or, rather, the lack thereof) of astrology, see
Section 3.5.
Section 2.3 The Moon, Our Constant Companion
This section discusses the Moon’s motion
and its observational consequences, including the lunar phases and eclipses.
• For what appears to be an easy concept, many
students find it remarkably difficult to understand the phases of the Moon. You
may want to do an in-class demonstration of phases by darkening the room, using
a lamp to represent the Sun, and giving each student a Styrofoam ball to
represent the Moon. If your lamp is bright enough, the students can remain in
their seats and watch the phases as they move the ball around their heads.
• Going along with the above note, it is
virtually impossible for students to understand phases from a flat figure on a
flat page in a book. Thus, we have opted to eliminate the “standard” Moon
phases figure that you’ll find in almost every other text, which shows the Moon
in eight different positions around Earth—students just don’t get it, and the
multiple moons confuse them. Instead, our Figure 2.22 shows how students can
conduct a demonstration that will help them understand the phases. The Phases
of the Moon tutorial on the MasteringAstronomy Web site has also proved very
successful at helping students understand phases.
• Note about the appearance
of lunar phases: We have often heard instructors describe the appearance
of the lunar phases in terms of, e.g., the illuminated portion of the moon
progressing from “right to left” during the cycle of phases. However, please
remember that this is true only of the Northern Hemisphere; it appears reversed
in the Southern Hemisphere, and in equatorial regions looks more like a bottom
to top. For that reason, we recommend not focusing on left/right and instead
focusing on time of visibility: waxing moons in the afternoon/evening and
waning moons in the morning.
• When covering the causes of eclipses, it
helps to demonstrate the Moon’s orbit. Keep a model “Sun” on a table in the
center of the lecture area; have your left fist represent Earth, and hold a
ball in the other hand to represent the Moon. Then you can show how the Moon
orbits your “fist” at an inclination to the ecliptic plane, explaining the
meaning of the nodes. You can also show eclipse seasons by demonstrating the
Moon’s orbit (with fixed nodes) as you walk around your model Sun: The students
will see that eclipses are possible only during two periods each year. If you
then add in precession of the nodes, students can see why eclipse seasons occur
slightly more often than every 6 months.
• The Moon Pond
painting in Figure 2.24 should also be an effective way to explain what we mean
by nodes of the Moon’s orbit.
• FYI: We’ve found that even many astronomers
are unfamiliar with the saros cycle of eclipses. Hopefully our discussion is
clear, but some additional information may help you as an instructor: The nodes
of the Moon’s orbit precess with an 18.6-year period; note that the close
correspondence of this number to the 18-year 11-day saros has no special
meaning (it essentially is a mathematical coincidence). The reason that the
same type of eclipse (e.g., partial versus total) does not recur in each cycle
is because the Moon’s line of apsides (i.e., a line connecting perigee and
apogee) also precesses—but with a different period (8.85 years).
• FYI: The actual saros period is 6585.32
days, which usually means 18 years, 11.32 days, but instead is 18 years 10.32
days if 5 leap years occur during this period.
Section 2.4 The Ancient Mystery of the Planets
This section covers the ancient mystery of
planetary motion, explaining the motion, how we now understand it, and how the
mystery helped lead to the development of modern science.
• We have chosen to refer to the westward
movement of planets in our sky as apparent
retrograde motion, in order to emphasize that planets only appear to go
backward but never really reverse their direction of travel in their orbits.
This makes it easy to use analogies—for example, when students try the
demonstration in Figure 2.33, they never say that their friend really moves
backward as they pass by, only that the friend appears to move backward against
the background.
• You should emphasize that apparent
retrograde motion of planets is noticeable only by comparing planetary positions
over many nights. In the past, we’ve found a tendency for students to
misinterpret diagrams of retrograde motion and thereby expect to see planets
moving about during the course of a single night.
• It is somewhat rare among astronomy texts to
introduce stellar parallax so early. However, it played such an important role
in the historical debate over a geocentric universe that we feel it must be
included at this point. Note that we do not give the
formula for finding stellar distances at this point; that comes in Chapter 15.
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 2.1
• (p. 27) The simple answer is no, because a
galaxy located in the direction of the galactic center will be obscured from
view by the dust and gas of the Milky Way. Note, however, that this question
can help you root out some student misconceptions. For example, some students
might wonder if you could see the galaxy “sticking up” above our own galaxy’s
disk—illustrating a misconception about how angular size declines with
distance. They might also wonder if a telescope would make a difference,
illustrating a misconception about telescopes being able to “see through”
things that our eyes cannot see through. Building on this idea, you can also
foreshadow later discussions of nonvisible light by pointing out that while no
telescope can help the problem in visible light, we can
penetrate the interstellar gas and dust in some other wavelengths.
• (p. 29) No. We can only describe angular
sizes and distances in the sky, so physical measurements do not make sense.
This is a difficult idea for many children to understand, but hopefully comes
easily for college students!
• (p. 30) Yes, because it is Earth’s rotation
that causes the rising and setting of all the objects in the sky. Note: Many instructors are surprised that this question often
gives students trouble, but the trouble arises from at least a couple
misconceptions harbored by many students. First, even though students can
recite the fact that the motion of the stars is caused by the rotation of
Earth, they haven’t always absorbed the idea and therefore don’t automatically
apply it to less familiar objects like galaxies. Second, many students have
trouble visualizing galaxies as fixed objects on the celestial sphere like
stars, perhaps because they try to see them as being “big” and therefore have
trouble fitting them onto the sphere in their minds. Thus, this simple question
can help you address these misconceptions and thereby make it easier for
students to continue their progress in the course.
• (p. 31 SIFY) This activity is designed to
help students become familiar with their local sky by learning their latitude
and then checking to see that the north or south celestial pole is indeed at
the altitude it should be.
• (p. 32 SIFY) This activity checks that
students can properly interpret Figure 2.14 and then asks that they go outside
to check their answers in the sky. Sample answer for September 21: The Sun
appears to be in Virgo, which means you’ll see the opposite zodiac
constellation—Pisces—on your horizon at midnight. After sunset, you’ll see
Libra setting in the western sky, since it is east of Virgo and therefore
follows it around the sky.
Section 2.2
• (p. 33) Jupiter does not have seasons because
of its lack of appreciable axis tilt. Saturn, with an axis tilt similar to
Earth, does have seasons.
• (p. 38) In 2000 years, the summer solstice
will have moved about the length of one constellation along the ecliptic. Since
the summer solstice was in Cancer a couple thousand years ago (as you can
remember from the Tropic of Cancer) and is in Gemini now, it will be in Taurus
in about 2000 years.
Section 2.3
• (p. 39 SIFY) This activity asks students to
observe the change in the Moon’s position among the stars over the course of
the night, making it another good way to help students connect their in-class
learning to the real sky.
• (p. 40) A “half light and half dark” moon
visible in the morning must be third-quarter, since third-quarter moon rises
around midnight and sets around noon.
• (p. 41) About 2 weeks each. Because the Moon
takes about a month to rotate, your “day” would last about a month. Thus, you’d
have about 2 weeks of daylight followed by about 2 weeks of darkness as you
watched Earth hanging in your sky and going through its cycle of phases.
• (p. 45) Remember that each eclipse season
lasts a few weeks. Thus, if the timing of the eclipse season is just right, it
is possible for two full moons to occur during the same eclipse season, giving
us two lunar eclipses just a month apart. In such cases at least one of the
eclipses will almost always be penumbral, because the penumbral shadow is much
larger than the umbral shadow and therefore it is more likely that the Moon
passes through it than through the smaller umbral shadow.
Section 2.4
• (p.
48) Opposite ends of Earth’s orbit are about 300 million kilometers apart, or
about 30 meters on the 1-to-10-billion scale used in Chapter 1. The nearest
stars are tens of trillions of kilometers away, or thousands of kilometers on
the
1-to-10-billion scale, and are typically the size of grapefruits or smaller. The challenge of detecting stellar parallax should now be clear.
1-to-10-billion scale, and are typically the size of grapefruits or smaller. The challenge of detecting stellar parallax should now be clear.
Solutions
to End-of-Chapter Problems (Chapter 2)
Visual Skills Check
1. B
2. D
3. C
4. d
5. b
6. d
7. c
8. c
Review Questions
1. A
constellation is a section of the sky, like a state within the United States Mediterranean , while the constellations of
the Southern Hemisphere got their official names from 17th-century Europeans.
2. If we were making a model of the celestial sphere
on a ball, we would definitely need to mark the north and south celestial
poles, which are the points directly above Earth’s poles. Halfway between the
two poles we would mark the great circle of the celestial equator, which is the
projection of Earth’s equator out into space. And we definitely would need to
mark the circle of the ecliptic, which is the path that the Sun appears to make
across the sky. Then we could add stars and borders of constellations.
3. No,
space is not really full of stars. Because the distance to the stars is very
large and because stars lie at different distances from Earth, stars are not
really crowded together.
4. The
local sky looks like a dome because we see half of the full celestial sphere at
any one time.
Horizon—The
boundary line dividing the ground and the sky.
Zenith—The
highest point in the sky, directly overhead.
We
can locate an object in the sky by specifying its altitude and its direction
along the horizon.
5. We
can measure only angular size or angular distance on the sky because we lack a
simple way to measure distance to objects just by looking at them. It is
therefore usually impossible to tell if we are looking at a smaller object
that’s near us or a more distant object that’s much larger.
Arcminutes
and arcseconds are subdivisions of degrees. There are 60 arcminutes in 1
degree, and there are 60 arcseconds in 1 arcminute.
6. Circumpolar
stars are stars that never appear to rise or set from a given location, but are
always visible on any clear night. From the North Pole, every visible star is
circumpolar, as all circle the horizon at constant altitudes. In contrast, a
much smaller portion of the sky is circumpolar from the United States
7. Latitude
measures angular distance north or south of Earth’s equator. Longitude measures
angular distance east or west of the Prime Meridian. The night sky changes with
latitude, because it changes the portion of the celestial sphere that can be
above your horizon at any time. The sky does not change with changing
longitude, however, because as Earth rotates, all points on the same latitude
line will come under the same set of stars, regardless of their longitude.
8. The
zodiac is the set of constellations in which the Sun can be found at some point
during the year. We see different parts of the zodiac at different times of the
year because the Sun is always somewhere in the zodiac and so we cannot see
that constellation at night at that time of the year.
9. If
Earth’s axis had no tilt, Earth would not have significant seasons because the
intensity of sunlight at any particular latitude would not vary with the time
of year.
10. The
summer solstice is the day when the Northern Hemisphere gets the most direct
sunlight and the Southern Hemisphere the least direct. Also, on the summer
solstice the Sun is as far north as it ever appears on the celestial sphere. On
the winter solstice, the situation is exactly reversed: The Sun appears as far
south as it will get in the year, and the Northern Hemisphere gets its least
direct sunlight while the Southern Hemisphere gets its most direct sunlight.
On the equinoxes, the two
hemispheres get the same amount of sunlight, and the day and night are the same
length (12 hours) in both hemispheres. The Sun is found directly overhead at
the equator on these days, and it rises due east and sets due west.
11. The
direction in which Earth’s rotation axis points in space changes slowly over
the centuries, and we call this change “precession.” Because of this movement,
the celestial poles and therefore the pole star changes slowly in time. So
while Polaris is the pole star now, in 13,000 years the star Vega will be the
pole star instead.
12. The
Moon’s phases start with the new phase when the Moon is nearest the Sun in our
sky; we cannot see the new moon, both because the Moon’s night side is facing
us and because the dim light we might otherwise see from the night side
(reflected light from Earth) is overwhelmed by the bright daytime sky. The
waxing phases — in which we see a gradually increasing amount of the Moon’s
visible face illuminated — then progress with one side of the Moon’s visible
face slowly becoming sunlit, moving to crescent, then to first-quarter (when we
see a half-lit moon), to gibbous and then to full. Full moon is when the entire
visible face of the Moon is sunlit and the Moon is visible all night long. The
waning phases then occur in reverse as the Moon’s sunlit fraction decreases,
through gibbous, third-quarter, crescent, and back to new again.
We can never see a full moon at noon because for the Moon
to be full, it and the Sun must be on opposite sides of Earth. So as the full
moon rises, the Sun must be setting and when the Moon is setting, the Sun is
rising. (Exception: At very high latitudes, there may
be times when the full moon is circumpolar, in which case it could be seen at
noon—but would still be 180° away from the Sun’s position.)
13. We
always see the same face of the Moon because the Moon displays synchronous
rotation, meaning that the Moon’s rotation period and its orbital period around
Earth are the same.
14. While
the Moon must be in its new phase for a solar eclipse or in its full phase for
a lunar eclipse, we do not see eclipses every month. This is because the Moon
usually passes to the north or south of the Sun during these times, because its
orbit is tilted relative to the ecliptic plane.
15. The
apparent retrograde motion of the planets refers to the planets’ behaviors when
they sometimes stop moving eastward relative to the stars and move westward for
a a few weeks or months. While the ancients had to resort to complex systems to
explain this behavior, our Sun-centered model makes this motion a natural
consequence of the fact that the different planets move at different speeds as
they go around the Sun. We see the planets appear to move backward because we
are overtaking them in our orbit (if they orbit farther from the Sun than
Earth) or they are overtaking us (if they orbit closer to the Sun than Earth).
16. Stellar
parallax is the apparent movement of some of the nearest stars relative to more
distant ones as Earth goes around the Sun. This is caused by our slightly
changing perspective on these stars through the year. The shift due to parallax
is very small because Earth’s orbit is much smaller than the distances to even
the closest stars. Because the effect is so small, the ancients were unable to
observe it. However, they correctly realized that if Earth is going around the
Sun, they should see stellar parallax. Since they could not see the stars
shift, they concluded that Earth does not move.
Does It Make Sense?
17. The constellation of Orion didn’t exist when my grandfather was a
child. This
statement does not make sense, because the constellations don’t appear to change on the
time scales of human lifetimes.
18. When I looked into the dark lanes of the Milky Way with my
binoculars, I saw what must have been a cluster of distant galaxies. This statement does not make
sense, because we cannot see through the band of light we call the Milky Way to
external galaxies; the dark fissure is gas and dust blocking our view.
19. Last night the Moon was so big that it stretched for a mile across
the sky. This
statement does not make sense, because a mile is a physical distance, and we
can measure only angular sizes or distances when we observe objects in the sky.
20. I live in the United States ,
and during my first trip to Argentina Argentina is in the Southern Hemisphere, the
constellations visible there include many that are not visible from the United States
21. Last night I saw Jupiter right in the middle of the Big Dipper.
(Hint: Is the Big Dipper part of the zodiac?) This statement does not make sense, because Jupiter,
like all the planets, is always found very close to the ecliptic in the sky.
The ecliptic passes through the constellations of the zodiac, so Jupiter can
appear to be only in one of the zodiac constellations—and the Big Dipper (part
of the constellation Ursa Major) is not part of the zodiac.
22. Last night I saw Mars move westward through the sky in its apparent
retrograde motion. This
statement does not make sense, because apparent retrograde motion is noticeable
only over many nights, not during a single night. (Earth’s rotation means that
all celestial objects, including Mars, move from east to west over the course
of each single night.)
23. Although all the known stars rise in the east and set in the west,
we might someday discover a star that will rise in the west and set in the east. This statement does not make
sense. The stars aren’t really moving around us; they only appear to rise in
the east and set in the west because Earth rotates.
24. If Earth’s orbit were a perfect circle, we would not have seasons.
This statement does not
make sense. As long as Earth still has its axis tilt, we’ll still have seasons.
25. Because of precession, someday it will be summer everywhere on Earth
at the same time. This statement
does not make sense. Precession does not change the tilt of the axis, only its
orientation in space. As long as the tilt remains, we will continue to have
opposite seasons in the two hemispheres.
26. This morning I saw the full moon setting at about the same time the
Sun was rising. This statement makes sense, because a full moon is
opposite the Sun in the sky.
Quick Quiz
27. c
28. a
29. a
30. a
31. a
32. b
33. b
34. b
35. a
36. b
Process of Science
37. (a)
Consistent with Earth-centered view, simply by having the stars rotate around
Earth. (b) Consistent with Earth-centered view by having the Sun actually move
slowly among the constellations on the path of the ecliptic, so that its position
north or south of the celestial equator is thought of as “real” rather than as
a consequence of the tilt of Earth’s axis. (c) Consistent with Earth-centered
view, since phases are caused by relative positions of Sun, Earth, and
Moon—which are about the same with either viewpoint, since the Moon really does
orbit Earth. (d) Consistent with Earth-centered view; as with (c), eclipses
depend only on the Sun-Earth-Moon geometry. (e) In terms of just having the
“heavens” revolve around Earth, apparent retrograde motion is inconsistent with
the Earth-centered view. However, this view was not immediately rejected
because the absence of parallax (and other beliefs) caused the ancients to go
to great lengths to find a way to preserve the Earth-centered system. As we’ll
see in the next chapter, Ptolemy succeeded well enough for the system to remain
in use for another 1500 years. Ultimately, however, the inconsistencies in
predictions of planetary motion led to the downfall of the Earth-centered
model.
38. The
shadow shapes are wrong. For example, during gibbous phase the dark portion of
the Moon has the shape of a crescent, and a round object could not cast a
shadow in that shape. You could also show that the crescent moon, for example,
is nearly between Earth and the Sun, so Earth can’t possibly cast a shadow on
it.
Group Work Exercise (no solution provided)
Short Answer/Essay Questions
40. The
planet will have seasons because of its axis tilt, even though its orbit is
circular. Because its 35° axis tilt is greater than Earth’s 23.5° axis tilt,
we’d expect this planet to have more extreme seasonal variations than Earth.
41. Answers
will vary with location; the following is a sample answer for Boulder , CO
a. The latitude in Boulder
b. The north celestial pole appears in Boulder
c. Polaris is circumpolar because it never
rises or sets in Boulder
42. a. When
you see a full Earth, people on Earth must have a new moon.
b. At full moon, you would see new Earth from
your home on the Moon. It would be daylight at your home, with the Sun on your
meridian and about a week until sunset.
c. When people on Earth see a waxing gibbous
moon, you would see a waning crescent Earth.
d. If you were on the Moon during a total lunar
eclipse (as seen from Earth), you would see a total eclipse of the Sun.
43. You
would not see the Moon go through phases if you were viewing it from the Sun.
You would always see the sunlit side of the Moon, so it would always be “full.”
In fact, the same would be true of Earth and all the other planets as well.
44. If
the Moon were twice as far from Earth, its angular size would be too small to
create a total solar eclipse. It would still be possible to have annular
eclipses, although the Moon would cover only a small portion of the solar disk.
45. If
Earth were smaller in size, solar eclipses would still occur in about the same
way, since they are determined by the Moon’s shadow on Earth.
46. This
is an observing project that will stretch over several weeks.
47. This
is a literary essay that requires reading the Mark Twain novel.
Quantitative Problems
48. a. There are 
arcminutes in a full
circle.
arcminutes in a full
circle.
b. There are 
arcseconds in a full
circle.
arcseconds in a full
circle.
c. The Moon’s angular size of 0.5° is
equivalent to 30 arcminutes or 
arcseconds.
arcseconds.
49. a. We know that 
so we can compute the
circumference of Earth:
so we can compute the
circumference of Earth:
b. There are 90° of latitude between the North
Pole and the equator. This distance is also one-quarter of Earth’s circumference.
Using the circumference from
part (a), this distance is
part (a), this distance is
So if 10,000 kilometers is the same as 90° of
latitude, then we can convert 1° into kilometers:
So 1° of latitude is the same as 111
kilometers on Earth.
c. There are 60 arcminutes in a degree. So we
can find how many arcminutes are in a quarter-circle:
Doing the same thing as in part (b):
Each arcminute of latitude represents 1.85
kilometers.
d. We cannot provide similar answers for
longitude, because lines of longitude get closer together as we near the poles,
eventually meeting at the poles themselves. So there is no single distance that
can represent 1° of longitude everywhere on Earth.
50. a. We start by recognizing that there are 24
whole degrees in this number. So we just need to convert the 0.3° into
arcminutes and arcseconds. So first converting to arcminutes:
Since there is no fractional part left to
convert into arcseconds, we are done. So 24.3° is the same as 
b. Leaving off the whole degree, we convert the
0.59° to arcminutes:
So we have 35 whole arcminutes and a
fractional part of 0.4 arcminute that we need to convert into arcseconds:
So 1.59° is the same as 
c. We have 0 whole degrees, so we convert the
fractional degree into arcminutes:
Since there is no fractional part to this, we
do not need any arcseconds to represent this number. So 0.1° is the same as 
d. We again have no whole degrees, so we start
by converting 0.01° to arcminutes:
There are no whole arcminutes here, either, so
we have to convert 0.6 arcminute into arcseconds:
So 0.01° is the same as 
e. We again have no whole degrees, so we start
by converting 0.001° to arcminutes:
There are no whole
arcminutes here, either, so we have to convert 0.06 arcminute into arcseconds:
So 0.01° is the
same as 
51. a. We will start by converting the 42
arcseconds into arcminutes:
So now we have 7°
38.7¢. Converting the 38.7 arcminutes to
degrees:
So 
¢ is the same as 7.645°.
¢ is the same as 7.645°.
b. We will start by converting the 54
arcseconds into arcminutes:
So now we have
12.9 arcminutes. Converting this to degrees:
So 
is the same as 0.215°.
is the same as 0.215°.
c. We will start by converting the 59
arcseconds into arcminutes:
So now we have 1°
59.9833¢ arcminutes. Converting this to
degrees:
So 1° 59¢ 59¢¢ is the same as 1.9997°, very close to 2°.
d. In this case, we need only convert 1
arcminute to degrees:
So 1¢ is the
same as 0.017°.
e. We can convert this from arcseconds to
degrees in one step since there are no arcminutes to add in:
So 1¢¢ is the
same as 
52. Answers
will vary for individual students based on size of their finger and arm length.
53. To
solve this problem, we turn to Mathematical Insight 2.1, where we learn that
the physical size of an object, its distance, and its angular size are related
by the equation:
We are told that
the Sun is 0.5° in angular diameter and is about 150,000,000 kilometers away.
So we put those values in:
For the values
given, we estimate the size to be about 1,310,000 kilometers. We are told that
the actual value is about 1,390,000 kilometers. The two values are pretty close
and the difference can be explained by the Sun’s actual diameter not being
exactly 0.5° and the distance to the Sun not being exactly 150,000,000
kilometers.
54. To
solve this problem, we use the equation relating distance, physical size, and
angular size given in Mathematical Insight 2.1:
In this case, we
are given the distance to Betelgeuse as 600 light-years and the angular size as
0.05 arcsecond. We have to convert this number to degrees (so that the units in
the numerator and denominator cancel), so:
We can leave the
distance in light-years for now. So we can calculate the size of Betelgeuse:
Clearly, we’ve
chosen to express this in the wrong units: lights-years are too large to be
convenient for expressing the size of stars. So we convert to kilometers using
the conversion factor found in Appendix A:
(Note that we could
have converted the distance to Betelgeuse to kilometers before we calculated
Betelgeuse’s size and gotten the diameter in kilometers out of our formula for
physical size.)
The diameter of
Betelgeuse is about 1.4 billion kilometers, which is more than 1000 times the
Sun’s diameter of 
kilometers. It is also
almost ten times the distance between Earth and Sun 
kilometers).
kilometers. It is also
almost ten times the distance between Earth and Sun kilometers).
55. a. Using the small-angle formula given in
Mathematical Insight 2.1, we know that:
We are given the physical size of the Moon
(3476 kilometers) and the minimum orbital distance (356,400 kilometers), so we
can compute the angular size:
When the Moon is at its most distant, it is
406,700 kilometers, so we can repeat the calculation for this distance:
The Moon’s angular diameter varies from 0.426°
to 0.559° (at its farthest point from Earth and at its closest, respectively).
b. We can do the same thing as in part (a),
except we use the Sun’s diameter (1,390,000 kilometers) and minimum and maximum
distances (147,500,000 kilometers and 152,600,000 kilometers) from Earth. At
its closest, the Sun’s angular diameter is:
At its farthest from Earth, the Sun’s angular
diameter is:
The Sun’s angular diameter varies from 0.522°
to 0.540°.
c. When both objects are at their maximum
distances from Earth, both objects appear with their smallest angular diameters.
At this time, the Sun’s angular diameter is 0.522° and the Moon’s angular
diameter is 0.426°. The Moon’s angular diameter under these conditions is
significantly smaller than the Sun’s, so it could not fully cover the Sun’s disk. Since it cannot completely cover
the Sun, there can be no total eclipse under these conditions. There can be
only an annular or partial eclipse under these conditions.
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