Building Models to Scale

![](media/image1.jpeg){width="5.758333333333334in"
height="2.327777777777778in"}

### 

### **Introduction**

Can you picture the dimensions of the solar system? Probably not. The
sizes and distances involved are so great that the mind tends to give
up, classifying the whole thing as "really, really big." It's even
difficult to represent on paper! Let me explain what I mean:

-   The above image shows the relative sizes of the Sun and her nine
    planets, but makes no attempt to represent the distances between
    them. Relative to their diameters, they're really much farther apart
    than shown.

![](media/image2.jpeg){width="3.0in" height="2.0in"}

-   The top diagram at right shows the true positions and orbital paths
    for the whole solar system as of August
    5^th^![](media/image3.jpeg){width="3.0in" height="2.0in"}, 2003. Can
    you tell how big each planet is from this view? What about the
    positions of the inner planets?

-   This next image "zooms in" on the previous one by a factor of 18,
    showing the state of the inner solar system on the same date. But
    now we've lost track of the outer planets!

You see? An 8.5" x 11" sheet of paper is just too small to represent
visually all of the sizes and separations in our solar system. What
about bigger paper? Well, blowing up the diagram of the outer solar
system to match the scale of the inner solar system diagram shown above
would require a poster 3 feet tall and 5 feet wide! And even then, the
sizes of the planets still would not be represented.

![](media/image4.jpeg){width="2.970138888888889in" height="1.2in"}So,
how can we visualize our solar neighborhood? One way is to build a
*scale model*. Take, for example, the model of a BMW Z4 Roadster shown
below. This model is advertised as "1/43 scale," which means that 1 inch
on the model represents 43 inches (a bit over 3.5 feet) on the real car.
This scale may also be written as a ratio, "1:43." In order to design
such a model, each and every feature of the real car must be measured
and then *scaled down* by this same factor of 43. The model car will
then be exactly 43 times smaller than the real thing, right down to the
last detail.

Today you will be building two models at very different scales. One will
feature the nine planets of our solar system, including their correct
and current positions as they orbit around the Sun. From this model you
will be able to predict what planets will be visible to us at night, and
you may even develop an observing schedule so that you won't miss any
opportunities to see them during an evening lab tonight. The other model
will be of the Earth-Moon system, which will allow us to see how human
beings compare to the vast sizes and distances of the Solar System. You
may be surprised to see just how far today's Space Shuttle astronauts
travel compared to the Apollo astronauts who once landed on the Moon!

Before we get started, answer the following questions in your lab
notebook. Don't worry about being right or wrong, just honestly record
your answers. You aren't expected to know everything yet, we're just
getting started!

1.  What real-world measurements do you think we will need to know in
    order to build our models?

2.  How much smaller than the real thing do you think a
    "reasonably-sized" model of the entire solar system would be? In
    other words, what scale should we use for our model? By
    "reasonably-sized," I mean that the planets should be large enough
    for us to see and manipulate, but the largest distances in the model
    shouldn't be so large that it becomes cumbersome to build and
    interact with the model.

![](media/image5.jpeg){width="3.125in"
height="2.0833333333333335in"}**Part I: The Solar System**

As I mentioned earlier, the first step to creating a scale model is to
measure all of the important properties of the full-sized object or
system. If you were around for our previous sessions, you know that
we've already taken care of this step. We used NASA's Solar System
Simulator website
([http://space.jpl.nasa.gov](http://space.jpl.nasa.gov/)) to look up the
distances of each planet from the Sun as of
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ on
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_, \_\_\_\_. This is the same
as \_\_\_\_\_\_\_\_\_\_\_\_\_

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ in Greenwich Mean Time (GMT,
sometimes also called Universal Time or UT), which is a standard time
that astronomers and other scientists can agree upon no matter where
they live. The image above is a sample output from the simulator for
9:30pm Central Daylight Time (CDT) on Monday, August 4^th^, 2003 (or
2:30am the next day in UT). As you can see from the image, the "range"
from the point of observation (the Sun) to Jupiter at that time was
803.009 million kilometers (803,009,000 km).

The list of these distances for all of the planets, as well as their
diameters, is given in Table 1 below. Notice that the planetary
diameters are "only" between a few thousand and a few hundred thousand
kilometers, while the distances range from 50 million to 5 *billion*
kilometers. In general, these clumps of gas and/or rock are tiny
compared to the distances that separate them!

+---------------+-------------+----------+------------------+---------+
| ## Celestial  | ## Diameter |          | ## Dista         |         |
|               |             |          | nce from the Sun |         |
+---------------+-------------+----------+------------------+---------+
| ***Body***    | ***(km)***  | ***(D~Ea | ***(km)***       | ***(A   |
|               |             | rth~)*** |                  | .U.)*** |
+---------------+-------------+----------+------------------+---------+
| # 0. Sun      | 1,392,000   | 109.13   | ~~\-\-\          | ~~\-\   |
|               |             |          | -\-\-\-\-\-\-\-\ | -\-\-\- |
|               |             |          | -\-\-\-\-\-\--~~ | \-\--~~ |
+---------------+-------------+----------+------------------+---------+
| **1.**        | 4,879       | 0.38     |                  |         |
| Mercury       |             |          |                  |         |
+---------------+-------------+----------+------------------+---------+
| **2.** Venus  | 12,104      | 0.95     |                  |         |
+---------------+-------------+----------+------------------+---------+
| **3.** Earth  | 12,756      | 1.00     |                  |         |
+---------------+-------------+----------+------------------+---------+
| **4.** Mars   | 6,794       | 0.53     |                  |         |
+---------------+-------------+----------+------------------+---------+
| **5.**        | 142,984     | 11.21    |                  |         |
| Jupiter       |             |          |                  |         |
+---------------+-------------+----------+------------------+---------+
| **6.** Saturn | 120,536     | 9.45     |                  |         |
+---------------+-------------+----------+------------------+---------+
| **7.** Uranus | 51,118      | 4.01     |                  |         |
+---------------+-------------+----------+------------------+---------+
| **8.**        | 49,528      | 3.88     |                  |         |
| Neptune       |             |          |                  |         |
+---------------+-------------+----------+------------------+---------+
| **9.** Pluto  | 2,390       | 0.19     |                  |         |
+---------------+-------------+----------+------------------+---------+

Table 1: Solar System Data

Because these numbers tend to be a bit overwhelming, we have also
converted them into some more convenient units. The unit of choice for
planet sizes is the *Earth-diameter*, which reveals that Jupiter's
diameter is about 10 times larger than the Earth's, while the Sun's is
about 10 times larger still. For distances between planets we use the
*astronomical unit* (*A.U.*), which is defined as the average distance
between the Earth and the Sun. All four of the inner, *terrestrial*
planets are less than about 1.5 A.U. from the Sun, while the farthest
*gas giant* is a whopping 30 A.U. away!

One last set of data that we need before we know how the real solar
system currently looks are the positions of the planets along their
orbits around the Sun. We obtained these angles from the Solar System
Simulator as well, using print-outs of the bird's-eye-view diagrams of
the outer and inner planets similar to those included on page 1 of this
lab. First we used a ruler to draw lines from the Sun to each of the
planets. We then used a protractor to measure the angles (centered at
the Sun) from the Sun-Earth line clockwise to each of the Sun-planet
lines we had drawn. This data is included in the "Position in Orbit"
column of Table 2 below.

Now that we know what the real solar system looks like on the night of
interest, we can consider what scale is appropriate for the construction
of our model. Our constraints are that we need to be able to fit the
model into the space we have available, that we have unobstructed
sight-lines from the Earth and Sun to all of the planets, and that the
planet-models themselves not be so small as to be invisible. Taking all
of these considerations into account, the best scale ratio seems to be
1:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_, or roughly
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ times smaller than the real thing. *One
meter in our model will represent* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
*kilometers out in space.* Using this scale, you calculated the scaled
diameters and distances for the model, and these are listed along with
the orbital positions in Table 2 below.

+-----------------+---------------+-------------------+---+-----------+
| ***Celestial*** | *             | ***Distance       | * |           |
|                 | **Diameter*** | from***           | * |           |
| ***Body***      |               |                   | * |           |
|                 | ***(mm)***    | ***the Sun (m)*** | P |           |
|                 |               |                   | o |           |
|                 |               |                   | s |           |
|                 |               |                   | i |           |
|                 |               |                   | t |           |
|                 |               |                   | i |           |
|                 |               |                   | o |           |
|                 |               |                   | n |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   |   |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | i |           |
|                 |               |                   | n |           |
|                 |               |                   | O |           |
|                 |               |                   | r |           |
|                 |               |                   | b |           |
|                 |               |                   | i |           |
|                 |               |                   | t |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
+-----------------+---------------+-------------------+---+-----------+
| # 0. Sun        |               | ~~\-\             | ~ |           |
|                 |               | -\-\-\-\-\-\-\-\- | ~ |           |
|                 |               | \-\-\-\-\-\-\--~~ | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | - |           |
|                 |               |                   | ~ |           |
|                 |               |                   | ~ |           |
+-----------------+---------------+-------------------+---+-----------+
| **1.** Mercury  |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **2.** Venus    |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **3.** Earth    |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **4.** Mars     |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **5.** Jupiter  |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **6.** Saturn   |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **7.** Uranus   |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **8.** Neptune  |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+
| **9.** Pluto    |               |                   |   | °         |
+-----------------+---------------+-------------------+---+-----------+

Table 2: Properties of the Model

We are finally ready to begin construction. We will need to make our own
super-long metric measuring tape, using a piece of string about 3
football-fields in length. We will be using printed images to represent
the planets, the correct scale for which is shown in the image below.
They will be affixed to placards on posts that we will plant at the
appropriate places out on the lawn. We'll use a 360˚ protractor,
straight-pins, and corkboard to construct all of our angles.

3.  As each new planet comes up for placement, ask yourself where you
    might expect it to be located based on what you've seen so far.

Your teacher will now give you instructions on how to lay our your
model. These instructions will be specific to the particular outdoor
space that you have available. You will need to find a location at which
to place your model Sun, and a direction from there towards your model
Earth. You will need to get both of these aspects right, so that you
will have access to every planet location (*e.g.*, they shouldn't be in
the woods or behind a fence), and so that you will be able to see all of
the planets from the models of both the Sun and the Earth.

Once the Sun and the Earth have been placed, you will use the model Sun
as your base of operations for placing the remaining planets. In order
to get the orbital positions right, you will use the Sun-Earth line as
the zero-line for angle measurements. Once the model is complete, you
will then move to the position of the model Earth, and again use the
Sun-Earth line to measure the angular positions of the planets as seen
in the night sky of the Earth. Your teacher will give you more detailed
instructions on how this will be done.

**Part II: Making Predictions**

Now that our model is complete, it's time to see what it can teach us.
By now you will have a true feel for the enormous distances separating
these relatively tiny planets. But even some of these "tiny planets" are
huge in their own right! (We don't call them "gas giants" for nothin'.)

This model can do much more than just make us say, "Wow." All of the
information we've put into it so far has been centered at the Sun, but
now we can walk over to the model Earth and see how the perspective
changes. Some of the planets are very close to us at the moment, so they
will be especially large and brilliant. Some others will be hiding
behind the Sun or too close to it to be seen against the glare. You
probably won't be able to see many of the model planets' disks with the
naked eye, but with binoculars or a telescope you can simulate what
you'll see during tonight's night-lab.

4.  The first step is to measure the angles -- centered at the Earth --
    that separate each planet from the Sun. Record this data in your lab
    notebook.

5.  From west to east (counterclockwise) and starting from the Sun, what
    is the order of the planets across the sky? This is the order in
    which they will rise and/or set.

![](media/image6.jpeg){width="2.529861111111111in"
height="2.1979166666666665in"}Next we will attempt to understand the
times at which each of the planets will rise and set tonight. Having
already measured all of the planetary angles as observed from the Earth,
we now have a handy pins-and-corkboard model of their locations in the
night sky (see Instructor Notes for more information). If we align the
180˚ mark on the protractor with the pin that represents the Sun, and
then mask off the bottom half of the protractor (180˚ - 360˚), we will
be able to represent the situation at sunset tonight. Every planet
between 0˚ and 180˚ on the protractor will be visible in the sky at
sunset, while all of those that are in the masked-off region will be
"below the horizon." Now we can rotate the protractor counter-clockwise,
simulating the rotation of the Earth. Notice that the Sun drops further
below the horizon as we do this, just like it will in reality! Some
planets will also set, while others will rise.

Now, the Earth rotates on its axis at a rate of 15˚ per hour, making a
full rotation in 24 hours. (24 hr x 15˚/hr = 360˚.) We will assume that
this constant rate applies to the apparent motion of the planets as
well, and from this assumption we will predict rough rising and setting
times for the planets. This is only an approximation, however, and due
to the fact that the plane of the solar system is not aligned to the
Earth's equator, these estimates could be wrong by as much as an hour.
(To really get this right, we would need to rotate the protractor at
different rates for different times of day.)

6.  What planets, if any, do you think are too close to the Sun to be
    visible to you against the glare?

7.  What planets, if any, do you think might be too far away from Earth
    to be visible?

8.  What planets do you think will be visible to us at sunset or within
    2 or 3 hours after?

9.  What planets do you think would be visible if you were up later than
    that, but before sunrise? If there are any planets in this category,
    at what time would you need to be awake to observe them?

Now, take a look at where the Moon is in the sky right now, and remember
where you've seen it over the past few nights. We will want to
incorporate it into our plans for the evening...

10. As a group, your final task will be to plan your observing schedule
    for this evening's night lab. Prioritize each planet based on when
    it will rise or set (you don't want to be looking for something that
    isn't up yet, or have something set before you've gotten to see it).
    If there are objects that will be up for the entire time, save them
    for when there are no other objects are available. We will have
    access to binoculars as well as a telescope.

11. Propose your observing schedule for approval by your instructors.

**Part III: Testing Predictions at Night**

12. For each object you try to observe, note the following information:

-   What equipment did you use?

-   If you couldn't find it, what do you think you would need to do for
    next time?

-   If you found it, was it where you expected?

-   What did you notice about the object?

13. Observe the Moon in binoculars or a telescope. What do you notice?
    How big do you think the features are that you see on the surface?

#### ![](media/image7.png){width="1.4076388888888889in" height="1.4354166666666666in"}Extension: The Earth and the Moon

Our previous scale model showed us that, although the Sun and the "gas
giant" planets are much larger than the Earth, the Solar System is
really mostly empty space. Now we will take a closer look at the Earth
and her partner the Moon, and discover just how we as human beings
compare in size to the vastness of the Solar System.

Most of us will spend our entire lives here on the planet Earth,
catching occasional glimpses of our companion the Moon. (If you *do* get
to go into space someday, send me a postcard!) Hundreds of astronauts
and cosmonauts have orbited the Earth, but only 12 have ever set foot on
the Moon, the last of whom were among the Apollo 17 crew back in 1972.

14. **Take a guess:** When Neil Armstrong became the first man to walk
    on the Moon back in 1969, how much farther from the surface of the
    Earth was he than one of today's shuttle astronauts on a servicing
    mission to the Hubble Space Telescope?

Now, let's begin to build a model of the Earth-Moon system. At this time
your group will receive a ball of clay from your instructor, which we
will divide up into models of the Earth and Moon.

15. **Take a guess:** How much of this clay do you think should be used
    to construct your model of the Moon, if the remainder must be used
    to make your model Earth? Record your answer. Divide the clay up
    according to your guess and note the relative sizes of your models.

Your instructor will now tell you the actual ratio of the volumes of
Earth and Moon. Remake your model according to this ratio.

16. How did your original model compare to this new model?

17. **Take a guess:** Based on intuition or previous knowledge, how far
    away do you think you should place the Moon from the Earth? Record
    your answer. Build your model. What was the reasoning behind your
    guess?

Now you will need the following facts: The diameter of the real Earth is
12,756 kilometers, and the true Earth-Moon separation is roughly 384,400
kilometers.

18. From the size of your model Earth and the diameter of the *real*
    Earth, use a calculator to determine the scale of your model. Then,
    calculate the appropriate separation between your Earth and Moon
    models. Record both answers. Correct your model.

19. How did your guess compare to the distance you've just calculated?

Now we will insert the "human perspective" that I mentioned above. When
a NASA space shuttle heads up on a servicing mission to the Hubble Space
Telescope, it orbits at a height of about 600 kilometers above the
surface of the Earth. This is the farthest above sea level that any
human has been since the last manned lunar mission back in 1972. We will
insert a toothpick into our model Earth, with the tip of the toothpick
representing the height of the shuttle orbit.

20. Using the scale of your model, calculate the height of the space
    shuttle orbit within the model. Record your answer. Add a toothpick
    to your model to represent this height, and compare this with the
    distance to the model Moon.

Would it be reasonable to build another full model of the Solar System
at this scale? To answer that question, consider what the Sun would look
like in this model. The real Sun is an average of 149,598,000 kilometers
from the Earth, and has a diameter of 1,392,000 kilometers.

21. Using the scale of your model, calculate the diameter of the model
    Sun you would need and the distance at which you would need to place
    it. Based on your calculations, do you think we could keep using
    this current scale to build another model of the whole Solar System?

**Appendix A: Answer Key and Examples**

1.  You will need to know the diameters of the planets, their distances
    from the Sun, & their orbital positions.

2.  The scale we used was 1:16,360,000,000. One meter in our model
    represented 16.36 million kilometers out in space, which is a scale
    of roughly 10 yards per astronomical unit.

-   Here is a sample version of Table 1:

+---------------+-------------+----------+------------------+---------+
| ## Celestial  | ## Diameter |          | ## Dista         |         |
|               |             |          | nce from the Sun |         |
+---------------+-------------+----------+------------------+---------+
| ***Body***    | ***(km)***  | ***(D~Ea | ***(km)***       | ***(A   |
|               |             | rth~)*** |                  | .U.)*** |
+---------------+-------------+----------+------------------+---------+
| # 0. Sun      | 1,392,000   | 109.13   | ~~\-\-\          | ~~\-\   |
|               |             |          | -\-\-\-\-\-\-\-\ | -\-\-\- |
|               |             |          | -\-\-\-\-\-\--~~ | \-\--~~ |
+---------------+-------------+----------+------------------+---------+
| **1.**        | 4,879       | 0.38     | 68,060,000       | 0.45    |
| Mercury       |             |          |                  |         |
+---------------+-------------+----------+------------------+---------+
| **2.** Venus  | 12,104      | 0.95     | 107,487,000      | 0.72    |
+---------------+-------------+----------+------------------+---------+
| **3.** Earth  | 12,756      | 1.00     | 151,768,000      | 1.01    |
+---------------+-------------+----------+------------------+---------+
| **4.** Mars   | 6,794       | 0.53     | 207,309,000      | 1.39    |
+---------------+-------------+----------+------------------+---------+
| **5.**        | 142,984     | 11.21    | 803,009,000      | 5.37    |
| Jupiter       |             |          |                  |         |
+---------------+-------------+----------+------------------+---------+
| **6.** Saturn | 120,536     | 9.45     | 1,351,000,000    | 9.03    |
+---------------+-------------+----------+------------------+---------+
| **7.** Uranus | 51,118      | 4.01     | 2,996,000,000    | 20.03   |
+---------------+-------------+----------+------------------+---------+
| **8.**        | 49,528      | 3.88     | 4,500,000,000    | 30.08   |
| Neptune       |             |          |                  |         |
+---------------+-------------+----------+------------------+---------+
| **9.** Pluto  | 2,390       | 0.19     | 4,589,000,000    | 30.68   |
+---------------+-------------+----------+------------------+---------+

Table 3: Solar System Data for Aug 5, 2003 at 2:30UT (*cf.* Table 1)

-   For the date of our model, note that the Earth was about 1% farther
    from the Sun than the average. The orbits of all of the planets are
    *elliptical*, with some of them being more non-circular than others.
    You may find it surprising that we are actually *farther away* from
    the Sun during a Chicago summer than in winter! It's not our
    proximity to the Sun that makes it hotter and colder, but the
    slanting rays of its light caused by the tilt of the Earth's axis.
    But that's a story for another day.

-   Here is a sample version of Table 2:

+-----------------+---------------+-------------------+---+-----------+
| ***Celestial*** | *             | ***Distance       | * |           |
|                 | **Diameter*** | from***           | * |           |
| ***Body***      |               |                   | * |           |
|                 | ***(mm)***    | ***the Sun (m)*** | P |           |
|                 |               |                   | o |           |
|                 |               |                   | s |           |
|                 |               |                   | i |           |
|                 |               |                   | t |           |
|                 |               |                   | i |           |
|                 |               |                   | o |           |
|                 |               |                   | n |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   |   |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | i |           |
|                 |               |                   | n |           |
|                 |               |                   | O |           |
|                 |               |                   | r |           |
|                 |               |                   | b |           |
|                 |               |                   | i |           |
|                 |               |                   | t |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
|                 |               |                   | * |           |
+-----------------+---------------+-------------------+---+-----------+
| # 0. Sun        | 85.09         | ~~\-\             | ~ |           |
|                 |               | -\-\-\-\-\-\-\-\- | ~ |           |
|                 |               | \-\-\-\-\-\-\--~~ | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | \ |           |
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|                 |               |                   | \ |           |
|                 |               |                   | - |           |
|                 |               |                   | - |           |
|                 |               |                   | ~ |           |
|                 |               |                   | ~ |           |
+-----------------+---------------+-------------------+---+-----------+
| **1.** Mercury  | 0.30          | 4.16              |   | 81.0°     |
+-----------------+---------------+-------------------+---+-----------+
| **2.** Venus    | 0.74          | 6.57              |   | 189.5°    |
+-----------------+---------------+-------------------+---+-----------+
| **3.** Earth    | 0.78          | 9.28              |   | 0.0°      |
+-----------------+---------------+-------------------+---+-----------+
| **4.** Mars     | 0.42          | 12.67             |   | 352.5°    |
+-----------------+---------------+-------------------+---+-----------+
| **5.** Jupiter  | 8.74          | 49.08             |   | 166.0°    |
+-----------------+---------------+-------------------+---+-----------+
| **6.** Saturn   | 7.37          | 82.58             |   | 218.5°    |
+-----------------+---------------+-------------------+---+-----------+
| **7.** Uranus   | 3.12          | 183.13            |   | 342.0°    |
+-----------------+---------------+-------------------+---+-----------+
| **8.** Neptune  | 3.03          | 275.06            |   | 1.0°      |
+-----------------+---------------+-------------------+---+-----------+
| **9.** Pluto    | 0.15          | 280.50            |   | 53.5°     |
+-----------------+---------------+-------------------+---+-----------+

Table 4: Properties of the Model for Aug 5, 2003 at 2:30UT (*cf.* Table
2)

3.  Student-specific response.

-   [Our example of how to begin the model:]{.underline} The first step
    is to place Uranus in its correct position, which is on the grass
    straight ahead of the Yerkes front steps. Facing the flagpole, we
    will then turn 30˚ east (clockwise) and use the model Sun-Uranus
    distance to find the correct location for the Sun. From here on out
    we will use the Sun as our base of operations, centering all of our
    angles there. Next up is to place the Earth 18˚ clockwise from the
    Sun-Uranus line. From now on, the Sun-Earth line will be our new
    zero-line for angle measurement. Next we will finish up the inner
    solar system with Mercury, Venus, and Mars, then work our way out to
    Jupiter, Saturn, and finally Neptune. Pluto will be left out of our
    model because it is inconveniently located on the neighboring golf
    course, but we will note the direction towards it from the Sun.
    (*Note: On the second day we rotated the model 7˚ counterclockwise
    about the Sun, in order to pull Neptune a bit farther away from the
    neighboring golf course. To facilitate this, we left a tent stake in
    the position of the Sun at the end of the first day.)*

-   When you're done building the model, you can tell your students
    where Alpha Centauri would be in the same model. In our case, the
    1.31-parsec distance to Alpha Centauri would have been represented
    by 2470 kilometers, or 1536 miles. This is roughly the distance from
    Yerkes Observatory to Helena, Montana!

4.  On Aug 5, 2003 at 2:30UT, the clockwise angles as viewed from the
    Earth were as follows:

> \| Sun = 0˚ \| Mercury = 334.5˚ \| Venus = 3.8˚ \| Earth = N/A \| Mars
> = 152.5˚ \|
>
> \| Jupiter = 347.1˚ \| Saturn = 34.4˚ \| Uranus = 160.7˚ \| Neptune =
> 180.5˚ \| Pluto = 235˚ \|

5.  On 8/5/03, the order from West to East was: Sun, Jupiter, Mercury,
    Pluto, Neptune, Uranus, Mars, Saturn, Venus. This is the order in
    which the planets will set in the West if they are already up at
    dusk, and the order in which they will rise in the East if they
    aren't.

6.  8/5/03: Venus, Jupiter.

7.  8/5/03: Pluto.

8.  8/5/03: Mercury, (Moon), Neptune, Uranus, Mars.

9.  8/5/03: Saturn.

10. 8/5/03: The planets should be prioritized in order from West to
    East. Check your local newspaper or an online resource to get the
    actual rise/set times for your area.

11. Student-specific response.

12. Student-specific response.

13. Student-specific response.

14. About 600x farther. Don't tell them yet!

15. Student-specific response.

-   The Earth:Moon volume ratio is about 49:1. One way for the students
    to get this proportion right is to roll the clay into a cylinder,
    then cut off 1/50^th^ of the length.

16. Student-specific response.

17. Student-specific response.

18. Using 25oz Play-Doh, the scale will be 1:115,820,000 or 1 cm = 1,160
    km. The models of Earth and Moon should be separated by about 3.3
    meters.

19. Student-specific response.

20. The space shuttle would only orbit 5mm above this model Earth!

21. The model Sun would be 12 meters in diameter, 1.3 km away. This
    would require 1.3 million times more clay!

**Appendix B: Instructor Preparation & Notes**

The first step to using this lesson with your students is to determine
what the Solar System will look like on the day that you want to build
your model. All of the information you will need has been made available
as a JavaScript application at:

<http://cfcpwork.uchicago.edu/cfcpdev/education/innercity/yerkes-summer2003.html#day>.

If you have access to binoculars and/or telescopes for use with your
students, the date and time you choose should correspond to the date and
time of their observing session. (*Note on UT:* Universal Time is
essentially the current time in Greenwich, England. In the USA, for
example, UT is 5 hours ahead of Central Daylight Time, and 6 hours ahead
of Central Standard Time.)

At first you should accept the default scale of 1:16,360,000,000. After
clicking the "Generate Values" button, the website will then tell you
the true diameters of the planets, as well as their current distances
from the Sun. Then, based on the default scale, the site will calculate
the diameters and distances that should be used within your scale model.
Finally, the website will calculate two sets of angles, both of which
apply to both the real Solar System and your model of it. The first set
gives the clockwise angle for each planet about the Sun, representing
that planet's position in its orbit. The second set are predictions of
the relative angular positions of the planets as viewed from the Earth,
which should match your students' observations when they stand at the
position of the model Earth.

![](media/image8.jpeg){width="3.473611111111111in"
height="3.473611111111111in"}Next you should ask yourself, "Do I have
enough space to build this model?" The model Sun-planet distances
themselves are not enough information to answer this question, as each
of these distances heads off in a different direction! You may find it
useful to look at the "Whole Solar System from 'Above'" image that can
be generated in Step 2, to give you a feel for the total space you will
need. As mentioned in the lesson plan above, your main goal should be to
have unobstructed sight-lines from the Earth and Sun to all of the
planets. Because the inner planets are relatively close to the Sun, you
should only need to consider the positions of the outer planets. Jupiter
completes one orbit in about 12 years, but Neptune takes 165 years to do
the same, and Pluto a whopping 249 years! That means that for the next
few years, the outermost of the outer planets will still be in roughly
the same orientation as they were in the image at right, which
represents our model for August 5^th^, 2003. (The large dome at Yerkes
Observatory measures 90 feet in diameter.)

Have you decided that you need to change the scale in order to fit the
model into your available space? If so, make sure that you regenerate
the table of values so that the 4^th^ and 5^th^ columns reflect the new
scale. The default scale of 1:16,360,000,000 was chosen because it
produces the smallest full model in which terrestrial planets like
Mercury and Pluto could still be represented to-scale (by small grains
of salt, or by a few pixels in an image printed at 300dpi).
High-resolution images at this scale are included in Appendix E below.
However, if you've decided to change the scale, you can use Step 3 of
the aforementioned website to generate low-resolution images of the
planets at the "correct" scale. Unfortunately, JPL's Solar System
Simulator (from which these images are taken) has a smallest size at
which it will print any of the planets, so the terrestrial-planet images
generated therein are actually larger than they should be.

![](media/image9.jpeg){width="2.0in" height="3.7430555555555554in"}

If you've decided to use images to represent the planets in your model,
one way to mount them is shown at right. For each planet, I used a
single self-adhesive laminating sheet to attach the image to foam-core
presentation board, which itself was duct-taped to a wooden dowel. We
had permission to stick these dowels into the lawn (pre-forming each
hole with a metal stake first), but if you don't have such permission
then you should look into some sort of a base for each of the planets.
The benefits of using images instead of little spheres to represent the
planets are:

-   You're less likely to lose the images.

-   It's impossible to glue down a grain of salt without the glob of
    glue becoming larger than the grain of salt.

-   Standing at the model Earth and observing the models of the other
    planets with binoculars or a telescope will yield the same view as
    when you observe the real planets with the same optics pointed at
    the night sky. A stable pair of binoculars shows that Saturn has
    rings!

How will you actually construct your model? Let's start first with the
distances. At the default scale, Neptune is separated from the Sun by
about 300 yards. Obviously, tape measures don't come in such lengths,
and certainly not marked with a metric scale. (I recommend sticking with
the metric system, because your students will need to calculate these
distances from the scale of the model, and you won't want to bog them
down with how to deal with decimal feet.) If you have the time to
position the planets one-after-another, I would recommend making a
single "metric tape measure" from a piece of string slightly longer than
the distance to the furthest planet in your model. You can start by
labeling the string at ten-meter intervals, with the intent to use meter
sticks to interpolate between these marks in the end. You wouldn't want
to do this along the entire length of the string, but notice that
Saturn, Uranus, and Neptune are within 10 meters of the 90m, 180m, and
270m marks, respectively.

![](media/image10.jpeg){width="2.0in"
height="2.2888888888888888in"}Tangling is a major issue with such a long
string, so be careful how you unwind it! A kite-string winder or
homemade equivalent is essential. For example, see the image at right of
our homemade winder in action. Features should include an axle that will
rotate freely in your hands, and perhaps a set of guards that will
prevent the string from coming off along the direction of the axle.

This string should be tied to a tent stake in the ground, or something
similar. On a table directly above that stake should sit a 10-inch
circular protractor on a square of corkboard. Anchor the protractor to
the cork with thumbtacks so that it's still free to rotate, then place
an orange straight-pin at the center to represent the Sun. Now, angles
will be measured by putting straight-pins representing the planets
around the perimeter of the protractor, as in the top picture at right.
We used the following color-code:

![](media/image6.jpeg){width="2.0in" height="1.7381944444444444in"}

> Sun - Orange
>
> Mercury - White
>
> Venus - Pink
>
> Earth - Green
>
> Mars -- Red
>
> Jupiter - Brown
>
> Saturn - Yellow
>
> Uranus - Blue
>
> Neptune - Purple
>
> Pluto - Black

![](media/image11.jpeg){width="2.0in" height="1.7354166666666666in"}The
bottom image at right shows how to get the planets placed at the right
angles out in the field. Once you have the Sun and Planet straight-pins
placed in the corkboard at the appropriate positions, you get down to
table-level and hide the Planet pin behind the Sun pin, with one eye
closed. Your task is then to have the Planet placard placed such that
it's hidden behind both of the above pins (at the correct distance, of
course).

Reading the geocentric angles out involves the opposite of the above
procedure. Move the table so that the center of the protractor sits
above the position of the Earth, and place a green straight-pin there to

represent the Earth. Now that you have all of the Planet placards in
their appropriate positions, get down to table-level and hide the
placard behind the Earth pin, and then place the Planet pin along the
perimeter of the protractor so that it's aligned with both of the above.

**Appendix C: Student Preparation (3 hours)**

To give the students full ownership of the model, we asked them to
research the current configuration of the solar system themselves. They
used NASA's Solar System Simulator website
([http://space.jpl.nasa.gov](http://space.jpl.nasa.gov/)) to determine
the distances from the Sun to each of the planets, and printouts from
this site along with rulers and protractors to find their angular
positions in their orbits. This data was then collected on an easel pad,
and the students themselves calculated the scaled-down distances to be
used in their models. These steps may be skipped in order to streamline
the whole exercise, but they add an element of inquiry to the lesson and
should be included if time and equipment allow it.

-   ![](media/image5.jpeg){width="3.140277777777778in"
    height="2.09375in"}Give students 5 minutes to experiment with the
    Solar System Simulator website.

-   Challenge them to find the distance from the Sun to their favorite
    planet at the given date and time. This is a good opportunity to
    explain how to convert from local time to Greenwich Mean Time, and
    why astronomers use GMT/UT.

    -   Answer: Ask the Simulator to show Planet X as seen from the Sun.
        The *Range* gives the distance between the two objects in
        **mil**lions or **bil**lions of kilometers.

-   Collect distances for all planets on an easel pad, upon which the
    diameters have been recorded in advance. Compare the distances
    (huge) with the diameters (relatively tiny!).

-   Ask the students to convert the diameter of their favorite planet
    into *Earth-diameters*, and its distance from the Sun into
    *astronomical units*. After you've collected the results on the
    easel pad, this will make the relative sizes and separations more
    comprehensible.

    -   1 A.U. = 149,598,000 km, the average distance between Earth and
        Sun.

    -   In the summer, you will find that the Earth is a bit farther
        from the Sun than average, and in the winter it is a bit closer.
        Many students will find this counterintuitive. This would be a
        good opportunity to discuss the reason for the seasons, if time
        allows.

-   ![](media/image2.jpeg){width="3.140277777777778in"
    height="2.09375in"}Challenge them to think of a way that they could
    use the above website to find the current orbital locations of the
    planets, *i.e.*, the angles that separate them from the Earth as
    viewed from the Sun.

    -   Answer: Ask the Simulator to show The Solar System as seen from
        Above. Fields of view of roughly 30˚ & 2˚ will show the outer
        and inner parts of the solar system, respectively. Printing them
        out and using protractors would provide the requested angles.
        But you will be prepared...

-   Provide them with printouts of the inner and outer solar system from
    the above website. (You will need to add a line to the latter
    representing the Sun-Earth direction.) Ask them to measure the
    angle, centered on the Sun, from the Earth *clockwise* to their
    favorite planet (0-360˚). Collect this data on a new easel sheet,
    which will contain the measurements necessary to build the model.

-   Give the students the scale of the model they are going to build,
    *i.e.*, conversion factors between *kilometers* in the real solar
    system and both *meters* (for distances) and *millimeters* (for
    diameters) in the model. Ask the students to calculate the diameter
    and separation from the Sun for their favorite planet in this model,
    and collect the data on the easel pad.

**Contents of Easel Sheets:**

-   Chart of True Sizes & Distances:

    -   Planet Name

    -   True Diameter (km & R~E~)

    -   True Distance (km & AU)

-   Chart of Model Specs:

    -   Planet Name

    -   Scaled Diameter (mm)

    -   Scaled Distance (m)

    -   Angle from Earth (deg CW)

**Appendix D: Student Results**

After three groups of students had built the same model on three
consecutive days, representatives from each group were brought together
to become "experts" at this lab. They performed some rudimentary data
analysis, in which they compared the model predictions from each of
their groups with each other and with the "professional predictions"
provided by the instructor. The data table they constructed is included
below as Table 5.

+--------------+-------------+-------------+-------------+-------------+
| ***          | ***Pre      | ***Pre      | ***Pre      | ***Prof     |
| Celestial*** | dictions*** | dictions*** | dictions*** | essional*** |
|              |             |             |             |             |
| ***Body***   | ***of Group | ***of Group | ***of Group | ***Pre      |
|              | 1***        | 2***        | 3***        | dictions*** |
+--------------+-------------+-------------+-------------+-------------+
| # 0. Sun     | 0˚          | 0˚          | 0˚          | 0˚          |
+--------------+-------------+-------------+-------------+-------------+
| **1.**       | 335.5˚      | 334˚        | 333˚        | 334.5˚      |
| Mercury      |             |             |             |             |
+--------------+-------------+-------------+-------------+-------------+
| **2.** Venus | 4.5˚        | 3.5˚        | 4.5˚        | **9˚**      |
+--------------+-------------+-------------+-------------+-------------+
| **3.** Earth | ~~\-\--~~   | ~~\-\--~~   | ~~\-\--~~   | ~~\-\--~~   |
+--------------+-------------+-------------+-------------+-------------+
| **4.** Mars  | 154.5˚      | 152.5˚      | 152˚        | 152˚        |
+--------------+-------------+-------------+-------------+-------------+
| **5.**       | 349.5˚      | 348.5˚      | **340˚**    | 347˚        |
| Jupiter      |             |             |             |             |
+--------------+-------------+-------------+-------------+-------------+
| **6.**       | 35.5˚       | 34.5˚       | 34.5˚       | 34.5˚       |
| Saturn       |             |             |             |             |
+--------------+-------------+-------------+-------------+-------------+
| **7.**       | 159˚        | 158.5˚      | 161˚        | 160.5˚      |
| Uranus       |             |             |             |             |
+--------------+-------------+-------------+-------------+-------------+
| **8.**       | 181˚        | 180˚        | 180˚        | 179.5˚      |
| Neptune      |             |             |             |             |
+--------------+-------------+-------------+-------------+-------------+
| **9.** Pluto | ~~\-\--~~   | ~~\-\--~~   | ~~\-\--~~   | 235.5˚      |
+--------------+-------------+-------------+-------------+-------------+

Table 5: Predictions of Planetary Positions in the Night Sky

First, the students used this data to improve their understanding of
concepts central to the lab. For example, they were able to explain to
their parents that the absolute agreement on the angular position of the
Sun was no coincidence, but rather that all four sets of predictions had
used the Sun as their zero-degree reference point. Likewise, they
understood that the lack of any data for the Earth was not a shortcoming
of their model, but rather the consequence of defining the Earth as our
observing platform, *i.e.*, the vertex of the angle defined above.
Finally, they knew that the lack of any student data for Pluto was
explained by its exclusion from their models, although they claimed
incorrectly that the reason for this was its great distance from the
Sun. In fact, Pluto would've been just 5 meters farther from the Sun
than was Neptune in our model, but it was its inconvenient placement --
through some trees, and in the middle of a nearby golf course -- that
led to its exclusion.

Next, in comparing the four sets of predictions, the students found
impressive agreement between them to within just a few degrees. However,
two exceptions were found, which are shown in bold in Table 3. First,
although the Jupiter data from Groups 1 & 2 agreed with each other and
with the professional predictions, Group 3's estimate was significantly
off. The students were quick to mock the individual(s) responsible for
this measurement, but they eventually came to understand that any number
of things could've gone wrong with that day's model and that it was
impossible to exactly specify the error. More significantly, they
noticed that the three student groups were in good agreement concerning
the position of Venus, but that they were all significantly short of the
given "professional" prediction. This led them to question the latter,
but the instructors again explained that pinpointing the source of the
error would require more research. The instructors then revealed that
the professionals had in fact been correct with an estimate of 4˚, but
that a typographical error had produced the value in question. This was
not intentional, but I would highly recommend *manufacturing* such an
error, as it gave the students a sense of pride by finding a flaw in
their instructor's work.

**Appendix E: High-Resolution Images**

The following images can be used to represent the Sun and planets at a
scale of 1:16,360,000,000.

![](media/image12.jpeg){width="6.0in" height="6.0in"}

![](media/image13.jpeg){width="2.0in" height="2.0in"}
![](media/image14.jpeg){width="2.0in" height="2.0in"}
![](media/image15.jpeg){width="2.0in" height="2.0in"}

![](media/image16.jpeg){width="2.0in" height="2.0in"}
![](media/image17.jpeg){width="2.0in" height="2.0in"}
![](media/image18.jpeg){width="2.0in" height="2.0in"}

![](media/image19.jpeg){width="2.0in" height="2.0in"}
![](media/image20.jpeg){width="2.0in" height="2.0in"}
![](media/image21.jpeg){width="2.0in" height="2.0in"}

Image Credits:

[http://space.jpl.nasa.gov](http://space.jpl.nasa.gov/)

<http://www.pbase.com/image/16572540>

<http://www.ewa1.com/md/min21030.jpg>

<http://www.nasm.edu/ceps/rpif/img/earth/earthrise.gif>

[http://www.mapquest.com](http://www.mapquest.com/)

Other Resources:

<http://www.fourmilab.ch/cgi-bin/uncgi/Solar> - See the Solar System
from above.

<http://www.fourmilab.ch/yoursky/> - See the night sky for any location
on Earth.