Module 24

1
Module 2-4
Scaling Galileos Solar System 4-Locating the
Globes
Where are the model planets on campus at various
times after the system is started? How far apart
are the model planets? How do the distances
between them vary with time?
Quantitative Concepts and Skills Scale Polar and
Cartesian coordinates Sine and cosine Degrees to
radians Pythagorean sum
2
Problem
With great fanfare, the University inaugurates
the model Galilean solar system. The President
of the University starts the planets in motion at
noon on Day Zero. At this starting time, all the
planets are lined up along the positive x-axis
(west to east). As shown on this map, Saturn is
on the eastern boundary of campus. Where
are all the planets at noon of the following day?
3
PREVIEW
Our data are (1) distance of the model planet
from the center of campus, and (2) the period of
its revolution. This period can be combined with
elapsed time to produce an angular measure of the
planets position relative to the x-axis. Thus
we have or can get the polar coordinates of the
planet radius and angle. To plot the position
of the planet on a campus map, we need only
convert the polar coordinates to Cartesian
coordinates. Slides 4-6 develop a spreadsheet
that converts polar coordinates of a single
planet to Cartesian coordinates and plots the
position of the planet on an XY (scatter) graph
that shows the outside boundaries of campus.
Slides 7-10 modify Slides 4-6 to show all of
Galileos planets on the campus map. The
planets positions are keyed to the number of
hours since the model was first turned on, so
that changing the time parameter from 24 to 48
hrs, for example, illustrates how the planets
move from the first day to the second day of the
models lifetime. Slide 11 describes how to
calculate the distance between two points from
their XY coordinates. Slide 12 expands the
spreadsheet of Slides 7-10 to include the
distance of each of the model planets from the
Earth at any user-selected time. Slides 13-14
call for a spreadsheet charting the distance of
model Venus and model Mars from model Earth day
by day for the first week of the models run.
4
Preliminary Laying out the map, and locating
points
Map of campus, showing outside square boundary
and location of an arbitrary position of Saturn.
Diagonal line to Saturn is the radius vector
horizontal and vertical lines to Saturn are x-
and y-projections, respectively. Map is an XY
(scatter graph) of Block C11D25. Skip rows to
break continuity of line (Rows 16, 17, 20, and
23).
Blocks C11D15, Cells C19 and D19 Cells C21,C22
and D22 and Cells C24, C25 and D25 all have
cell formulas referring back to user-defined
numbers in C4, C6 and C7. As you change the
angle in C6, the position of Saturn should
change.
Duplicate this spreadsheet including the graph
and study how the location of Saturn changes as
the angle (C6) changes. See next two slides for
some examples.
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Preliminary Laying out the map, and locating
points, 2
Angle (Cell C6) 120 degrees
Angle (Cell C6) 205 degrees
6
Preliminary Laying out the map, and locating
points, 3
Angle (Cell C6) 340 degrees
Angle (Cell C6) 523 degrees Note, this is
523/360 cycles.
7
Cells I14J19, the targets of this spreadsheet,
are the Cartesian coordinates of the sun and
planets 24 hours after the start of the working
model. Cells D14D19 and G1419 are the polar
coordinates.
Locating the planets on campus
Cells D4 and D5 are scale factors from previous
modules. D7 is the key changeable parameter.
D13D19 and F14F19 are from Columns C and E and
the scales. Col G is from D7 and Col F. H is
from G. I and J are from D13D19 and G14G19.
Duplicate this spreadsheet and add a map
positioning the sun and all of Galileos planets
on campus. Use an XY (scatter) graph for Cells
I13J19. Add corner boundary points and connect
them as in Slide 4 so that you can form the map
area into a square simply by pulling on the
handles.
8
Locating the planets on campus, 2
After 1 day
Skip a column so you can format the data series
in Column K (lines, no markers) differently from
the data series in J (markers, no lines).
Hard to see the first four planets. So add a
second map (enlargement) with edges 0.15 km from
the sun.
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Locating the planets on campus, 3
After 1 day
Study how the location of the planets change by
changing D7. For example, these maps show where
the planets are one day after start of the model.
Where are they after two days?
10
Locating the planets on campus, 4
Changing this number changes all the numbers in
G14J19 as well as the two maps
After 2 days
Which of the planets is Earth? Suppose you were
standing on the night side of Earth. Would you
be able to see any of Galileos planets in your
night-time sky?
11
Distance between planets
One of the reasons that the alumni want this
working scale model is that it will show how the
distances between planets vary as they revolve
around the sun.
Key to the following spreadsheets
You can easily calculate the distance between two
points if you know their Cartesian coordinates.
Coordinates x1, y1
Distance SQRT((x1-x2)2 (y1-y2)2)
Sometimes called the Pythagorean sum.
Coordinates x2, y2
12
Distance between planets, 2
Retrieve your spreadsheet from Slide 7 and add
Column L for the distance between each of the
model planets and the model Earth.
One day after the start of the model, Venus and
Mars are almost exactly the same distance from
the Earth. Can you see that in the map of Slide
9?
Now construct a completely new spreadsheet
showing how the distances of Venus and Mars from
Earth in the model vary from day to day for the
first week after the model is turned on.
13
Distance between planets, 3
Rows 6-8 and 10-12 are from Columns D and F of
Slide 7. Rows 14-16, 18-20, and 22-24 are
calculated using formulas used in Slide 7. Rows
26 and 27 are the Pythagorean sums from Rows
18-20 and 22-24.
Here are the bottom lines distances of Venus and
Mars at noon of the first seven days. Add a
graph to your spreadsheet showing these distances
vs. days in Row 3.
14
Distance between planets, 4
It looks like Mars gets further away from Earth
than Venus does. Is this reasonable? What is
the maximum distance of these planets from Earth?
What is the minimum?
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End of Module Assignments

• Answer the questions in Slides10 and 14.
• Use your answers to the question in Slide 14 to
  draw an envelope (a horizontal line for the
  maximum, and another for the minimum) for the
  distances plotted on the graph in that slide.
  Add enough intermediate times so that the shape
  of the curves becomes evident. In particular,
  find the time of the maximum and minimum. (The
  curves in Slide 14 suffer because of an
  insufficiency of data points. What do the curves
  really look like?)
• How large would the square campus have to be to
  contain Uranus, Neptune, Pluto? Expand the map
  of Slide 8 to that size and include the three
  newer planets. Retain the boundary of the
  present campus.
• Add Column M to the spreadsheet in Slide 12 to
  give the direction (measured from the x-axis)
  from Earth to the sun and each of the planets.
  For direction, use the angle from the x-axis as
  calculated by the arctangent function. Print
  results in degrees. Compare your results to the
  map in Slide 9. Are your results reasonable?
• (Continuation of assignment 4) Columns L and M
  give the position of the sun and planets relative
  to the Earth in polar coordinates. Draw a map
  showing these relative positions, as if the Earth
  were at the center of the universe, as it was
  before Copernicus.