This is a lab used in Introductory Astronomy courses at the University of Washington. The original conception is by Dr. Bruce Balick, with modifications by Dr. Woody Sullivan and subsequently by Dr. Doug Ingram. It is originally modelled on a lab using the Voyager software for the Mac. In that version, the students took the data themselves, using the program to guide them.

There is a long section that uses the motion of the moon in the sky to take the student on a virtual tour of the solar system. Of course, the actual dates and objects in the sky will depend upon the time, so this lab must be updated each time it is used!

Introduction

We have all viewed and enjoyed the Moon in its many phases from that of a fingernail crescent to the full moon. We all know that the bright part of the Moon is illuminated by the Sun and that the phases we see are somehow related to the locations of the Sun and Moon relative to the Earth. We explore these geometric relations in this exercise. Interestingly, a poll of Harvard graduates revealed that most of them believe the dark part of the Moon is the result of portions of the Moon lying in the shadow of the Earth. Their opinion is ingenious, but dead wrong. We'll find out why.

The specific goals of the lab are to understand how the various phases of the Moon arise and why the phases are related to the differences of rising and setting times of the Moon and Sun. Be sure to read and understand pp. 27-31 in the text before starting the lab.

Observations

Here are some basic observations of the night sky as a starting point.

• The phase of the Moon changes over the course of one month, from new, to waxing crescent, to first quarter, etc., and back to new.

• Over the course of a couple of weeks, as the Moon moves closer in the sky to the Sun (from its position at full moon moving to new moon), it gets darker as a larger and larger fraction of it is no longer illuminated.

• Likewise, as the Moon moves away in the sky from the Sun, it gets brighter.

• The shape of the dark region on the Moon is like a circular cut out of the Moon (similar to what the Earth's shadow would do) but only when the Moon is near its new moon phase.

Let's make a detailed table of observations by following the Moon in the night sky over the course of one month, starting from March 27, 1994 at 6 pm Pacific Standard Time in Seattle and taking data in intervals of roughly 48 hours. We'll record the date, the phase of the Moon, the fraction of the Moon that is illuminated, the angle between the Sun and the Moon, and the angular size of the Moon (in arcminutes, where 60 arcminutes is one degree on the sky). To save us all some time, the data we'd collect is reproduced here.

--------------------------------------------------------------
|Date  |Phase            |%Illum  |Moon-Sun Angle  |Moon Size|
|------|-----------------|--------|----------------|---------|
|3/27  |Full             |100     |172             |32.9'    |
|3/29  |Waning Gibbous   | 95     |155             |33.2'    |
|3/31  |Waning Gibbous   | 81     |128             |32.4'    |
|4/2   |Waning Gibbous   | 60     |102             |31.4'    |
|4/4   |Waning Crescent  | 40     | 78             |30.5'    |
|4/6   |Waning Crescent  | 22     | 55             |29.9'    |
|4/8   |Waning Crescent  |  4     | 22             |29.5'    |
|4/10  |New              |  0     |  2             |29.5'    |
|4/12  |Waxing Crescent  |  4     | 21             |29.6'    |
|4/14  |Waxing Crescent  | 14     | 43             |30.0'    |
|4/16  |Waxing Crescent  | 30     | 66             |30.6'    |
|4/18  |1st Quarter      | 50     | 90             |31.4'    |
|4/20  |Waxing Gibbous   | 71     |115             |32.3'    |
|4/22  |Waxing Gibbous   | 89     |142             |33.1'    |
|4/24  |Waxing Gibbous   | 99     |169             |33.4'    |
|4/26  |Full             |100     |175             |33.3'    |
--------------------------------------------------------------

If you were to build this table of observations yourself, you'd notice over the course of the month, the Moon takes you on sort of a ``guided tour'' of the solar system. Here are some of the high points for this particular month:

• March 30 -- Moonrise is 10:08 pm, a few hours after sunset. Jupiter can be found about 7 degrees above the Moon, which is in the Southeastern sky. With a good telescope, Pluto can be found about 18 degrees to the left of the Moon on the sky.

• April 4 -- The Moon is up in the morning in the Southwestern sky now, and with a good telescope, you could see Uranus and Neptune very close to one another in the sky, only about 5 degrees to the left of the Waning Crescent Moon.

• April 7 -- The Moon is still in Waning Crescent phase and is up in the morning in the Southeast. Just before dawn, at about 5:15 am, you can find three planets near the Moon: Saturn is down and to the right about 6 degrees, while Mercury and Mars are paired in the sky about 15 degrees down and to the left of the Moon, right on the horizon, with Mercury the dimmer and lower altitude of the two planets.

• April 12 -- The Waxing Crescent Moon almost eclipses Venus at about 3:30 pm standard time today, coming within half a degree! By sunset, you can easily see Venus about a degree above the Moon in the Southwestern sky.

• April 14 -- Toward the Southwest in the evening, you may notice a bright star about 4 degrees to the left of the Waxing Crescent Moon. It's Aldebaran, the brightest star in Taurus.

• April 17 -- If you can find the Moon during the evening, it will be almost right in between two very bright stars (13 degrees above and below the Moon), Pollux (above) and Procyon (below).

• April 24 -- The Moon is rising just before sunset now, and if you look just above the Moon, you'll notice it is very close to eclipsing the bright star Spica (in Virgo).

• April 26 -- By now, the nearly Full Moon has come full circle in the sky and is once again found close in the sky to Jupiter, rising near sunset with Jupiter only a few degrees to the left on the sky.

The appearance of all these objects close to the Moon is not the only surprise in the data. Notice the angular size of the whole Moon is changing with time (not just the illuminated part...the whole thing!). Think about the possibility of a correlation here between the angular size of the Moon and any other things, like illumination or Moon-Sun angle. Notice how the Moon-Sun angle seems to rise and fall with the illuminated fraction of the Moon. Think about what the Moon-Sun angle means and whether or not this apparent correlation matches what we guessed earlier in our list of basic observations.

Analysis

Here's where you get to start doing the work. Answer all of the questions given below on your own paper. Please use graph paper for any plots you are asked to make, simply to make the graphs easier for you to read and understand.

The general idea is to determine whether the observations can be successfully interpreted in terms of a model--in this case, the model described in Chapter 2 in your text. Note that this model predicts which lunar phases a terrestrial observer sees based on certain assumptions about the Earth-Moon-Sun geometry. Your job is to see if the model is fully consistent with your observations. Of course, all of the phases of the Moon are observed as predicted (otherwise this model would never have been proposed in the first place!!). But now, let's take a ``God's-eye view'' of what causes the lunar phases.

Chapter 2 and figure 2-15 tell us that the angle between two lines, one from the Earth to the Sun, and a second from the Earth to the Moon, completely determines the lunar phase an observer is expected to see. We have already measured that angle as well as the lunar phase, so the analysis should be straightforward.

• (1) (14 points) Draw a diagram describing where you think the Moon is with respect to the Sun in each of its 8 major phases (ask for help if you are unsure how to do this) and derive an expected relationship between the Sun-Moon angle and the observed lunar phase (this can be shown by labelling your diagram with phases and approximate Sun-Earth-Moon angles).

• (2) (14 points) There are other models besides the one in the book that might be consistent with the observations. Let's test one of them. This is the ``Harvard Hypothesis'' in which the shadow of the Earth falls onto a portion of the Moon, causing it to appear dark. The remaining sunlit portions are then visible, and they supposedly account for the phases of the Moon that we observe. Following the same procedure as in (1), draw the Earth's shadow and see whether this hypothesis is consistent with the observed data by taking a couple of sample point off the table and seeing if they match up with your diagram. You should only need to compare for two or three points.

Some of the questions which follow may require you to refer to your text or lecture notes.

• (3) (14 points) Plot a graph showing the Illuminated Fraction vs the Moon-Sun angle. We say that a correlation exists between two things if they seem to behave the same way in a graph. When a correlation is found, we start looking for a reason behind it (think about this in the context of the Pine reading discussion on correlation, causation and scientific studies). Does a correlation exist here?

• (4) (10 points) Note from the table that the angular size seems to change over the course of a month. We know that the Moon appears larger near the horizon though it actually isn't (this is an optical illusion... check it yourself with a measuring device!). Why does the angular size of the Moon (not just the illuminated part...the whole thing) change over the course of a month (this variation is real)? Think about this for a while before asking someone for help...the most important step in science is that intuitive leap you make on your own.

• (5) (10 points) Is the angular size of the whole Moon correlated with anything else in the table, and is the correlation showing you the cause of the effect (or is it just accidental)?

• (6) (14 points) Another interesting fact we can gather from our observations is that the Moon seems to pass close in the sky to every single planet at some point or other during its orbit. This is no coincidence! What is it about the orbit of the Moon with respect to the orbits of the other planets that causes this? Again, think about this on your own for a while at least! Ask yourself why you never see the Moon next to, say, Polaris (the star marking the North Celestial Pole)...the Moon always seems to be in the Southern sky when viewed from Seattle.

  (7) (24 points) Looking back at the diagram we drew in problem (1), it looks as if every time we're in the full moon phase, the Earth is interposed between the Sun and the Moon. Yet eclipses of the Moon (in which the shadow of the Earth partially or totally obscures the Moon) happen much less often than once per month (more like once or twice a year). So why doesn't the Moon always get eclipsed to some degree during its full phase? The answer to this question also explains why we don't have solar eclipses every month! Draw a diagram to help explain your answer.