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PHYS 175, FALL 2014
HW #1 S OLUTION
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1.10 What is the difference between a solar system and a galaxy?
A solar system comprises one or two (perhaps even three) stars, about which planets (terrestrial or gaseous) orbit. A galaxy is made up of millions or even billions of stars. Galaxies may
contain many millions of solar systems, all orbiting a common center of mass. In terms of
scale size, a solar system spans of order 10-100 AU, whereas a galaxy is measured in terms of
light years (1 ly ≈ 63,000 AU). In common, both are consequences of gravitational attraction
of matter resulting in orbital motion.
1.23 What is the meaning of the letters R I V U X G that appear with some of the figures in this
chapter? Why in each case is one of the letters highlighted?
These letters represent a mnemonic for the electromagnetic spectrum: Radio Infrared Visible
Ultraviolet X-ray Gamma ray. This is similar to the familiar R O Y G B I V used for the visible portion of the spectrum. Astronomers use light to observe the universe, but not all of
that light is visible – hence the electromagnetic spectrum, which describes electromagnetic
waves based on their energy. The highlighted letter indicates which band within the electromagnetic spectrum the image was created with. For anything other than the visible range,
astronomers map energies to visible colors in what is called a ”false color” image.
1.24 The diameter of the Sun is 1.4 × 1011 cm, and the distance to the nearest star. Proxima
Centauri, is 4.2 ly. Suppose you want to build an exact scale model of the Sun and Proxima
Centauri, and you are using a ball 30 cm in diameter to represent the Sun. In your scale
model, how far away would Proxima Centauri be from the Sun? Give your answer in
kilometers, using powers-of-ten notation.
The easiest way to do this problem is to set a scale and use dimensional analysis. Below is
one method to accomplish the task.
30 cm
9.46 × 1012 km
≈ 8, 510 km = 8.51 × 103 km
·
4.2
ly
·
1 ly
1.4 × 1011 cm
Note that dimensions (units) cancel in fractions just like numbers, so the scale is a dimensionless cm/cm, the light years cancel out as well, leaving only km. So for a 30-cm Sun that’s
only a bit bigger than a regulation basketball (which is about 24 cm in diameter), you’d have
to place the model for Proxima Centauri some 8,500 km (roughly 5,300 miles) away! If that
Sun were in Fairbanks, Proxima Centauri’s facsimile would be in Konya, Turkey.
2.19 At what point on the horizon does the vernal equinox rise? Where on the horizon does it
set?
Looking at Fig. 2-16 in the text, the vernal equinox sunrise is directly in the east. Similarly, it
would set almost directly in the west.
2.26 What is the difference between the sidereal year and the tropical year? Why are the two
kinds of year slightly different in length? Why are calendars based on the tropical year?
The sidereal year is based on the distant stars, whereas the tropical year is based on the
Sun. The former is useful for astronomical observations and predictions about the stars and
PHYS 175, FALL 2014
HW #1 S OLUTION
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galaxies. The difference in length of time for each type of year is easiest explained by looking
at a single day. For a point on Earth go from one solar noon to another, the planet must rotate
through slightly more than 360◦ (this is since the Earth is simultaneously orbiting the Sun) –
this is a tropical day. A sidereal day requires only 360◦ rotation, so it will be shorter than a
tropical day. We use the tropical year for our calendars so that the seasons occur in the same
months from year to year.
2.43 Ancient records show that 2000 years ago, the stars of the constellation Crux (the Southern
Cross) were visible in the southern sky from Greece. Today, however, these stars cannot
be seen from Greece. What accounts for this change?
Much like a top spinning, the Earth’s rotational axis precesses in a circular path. This changes
the celestial equator and therefore what stars can and cannot be seen from a particular latitude. We don’t notice this change to much, since it takes roughly 26,000 years for the Earth to
completely precess once. This precession also explains why Polaris isn’t always our ”North
Star.”
3.4 At approximately what time does the Moon rise when it is (a) a new moon; (b) a first
quarter moon; (c) a full moon; (d) a third quarter moon?
An easy way to remember how this works is to recall that the Sun illuminates the Moon.
So when the Moon is full (we see it’s bright half facing us) it must be that the Sun is in the
opposite part of the sky. So, (a) a new moon will rise roughly at sunrise – the Sun must be
”behind it” from our point of view, so we see the shadowed portion. (b) A first quarter moon
will rise about 6 hours after sunrise. (c) The full moon will rise at about sunset. (d) The third
quarter moon will rise roughly 6 hours after sunset.
3.24 How did Aristarchus try to estimate the diameters of the Sun and Moon?
He used geometry and proportions to estimate the relative distances between the Sun, Earth
and Moon. Once he had these, he used the eclipses to state that the Sun and Moon had
the same angular size, so their relative sizes could be estimated in proportion to their relative distances. Although Aristarchus did not do it himself, once the result of Eratosthenes’
estimate of Earth’s diameter was known, other astronomers from the Alexandrian school
could estimate the diameter of the Sun and Moon. It turns out they were not too far in error,
especially for measurements taken more than 2000 years ago!
3.37 One definition of a ”blue moon” is the second full moon within the same calendar month.
There is usually only one full moon within a calendar month, so the phrase ”once in a blue
moon” means ”hardly ever.” Why are blue moons so rare? Are there any months of the
year in which it would be impossible to have two full moons? Explain your answer.
From the perspective of an observer on Earth, the lunar cycle is described by a synodic
month, and this takes about 29.5 days. Since most months have 30 or 31 days, that requires a
full moon to occur in the first day or two of the month for a blue moon to occur by month’s
end – this explains the relative scarcity of blue moons. February, which is, at most 29 days
long, cannot be host to a blue moon.
PHYS 175, FALL 2014
HW #1 S OLUTION
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3.46 Describe the cycle of lunar phases that would be observed if the Moon moved around
Earth in an orbit perpendicular to the plane of Earth’s orbit. Would it be possible for both
solar and lunar eclipses to occur under these circumstances? Explain your reasoning.
First, some similarities. We can assume that we would still only see one face of the Moon,
since it would still be phase-locked in it’s orbit around the Earth, as it is now. The lunar
phases would now have a seasonal aspect to them. Let’s start with the assumption that
the Moon’s orbital plane is perpendicular to the Earth-Sun line in January. For half the
lunar month we’d see only quarter moons, as in the figure, below. Then the Moon would
disappear for 14 days, below the horizon. Three months later, that orbital plane would be
parallel to the Earth-Sun line in April, and then we’d see a 14-day progression of phases
from full to new moons. Then cycle would repeat and by October we’d see a new-to-full
progression. In this scenario, we’d see occasional lunar eclipses during the beginning of
March or middle of October. Similarly, we’d see potential solar eclipses occurring in either
mid-March of early October. We’d see that eclipses were much rarer than before, since they
can only occur in those relatively short windows of time, rather than at any time of year.