EXPLORING THE MATHEMATICS OF THE SOLAR SYSTEM AND GPS
Our Solar System: The arrangement of nine planets around our comfortable sun offers
unique opportunities for exploring mathematics and science concepts.
Mathematics and Science topic areas:
Astronomy, Measurement, Geometry, Statistics,
1. We can look at the relationship that exists between the average Astronomical
Units (AU) the planet is from the Sun vs. that planet’s period of revolution in
Earth Years.
An Excel Spreadsheet or TI-83+ Lists would be useful.
Example with Excel – www.mste.uiuc.edu/dildine/times/planets.xls:
Planet
AU
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Years
0.39
0.72
1.00
1.52
5.20
9.55
19.17
30.06
39.44
0.24
0.62
1.00
1.88
11.87
29.48
84.07
164.90
248.71
Period of Rev in Earth Years
Orbital Distance vs. Earth Years (pd of revolution)
300.00
250.00
200.00
150.00
100.00
50.00
0.00
0.00
10.00
20.00
30.00
40.00
50.00
Average Orbital Disance (AU)
Students can and should explore the relationship that this graph represents.
2. We can also look at the relative sizes of the planets. Imagine that our Sun is only
9 inches in diameter. What sizes would the rest of the planets have to be? Here is
a snapshot of a Web Site that allows us to explore that very question. This is
located at http://www.exploratorium.edu/ronh/solar_system/
3.
You can use this site to manipulate the different values for the Sun. What size is the
Sun if the Earth is the size of a regulation basketball (12 inches in diameter)?
What size is Jupiter in comparison to the Earth of that size?
4. We can also look at the average orbital distance of each planet when we set the
sun to be a particular size.
A Mile is 5,280 feet or 1,600 meters. About how far would our model planet Pluto be
from the Sun when the Sun is 6 inches is diameter? What about if we set the Sun to be
3000 inches across?
GPS receivers take in data from the Global Positioning System, a constellation of 24 - 27
satellites orbiting the earth. The satellites were developed and launched by the United
States Department of Defense at a cost of more than $12 billion. This system is the most
advanced navigational technology ever developed. It can provide your precise latitude
and longitude, altitude, and speed; the direction in which you are moving; and the time
anywhere on Earth, in any weather, at any time. Because a GPS receiver must track data
from at least three satellites at once (4 for precise elevation), it must have a direct line of
sight to the sky at any given time you are using it.
The GPS Units can also function as a highly accurate measuring device (kind-of an
electronic tape measure).
As we have seen there are several numerical relationships between the planets in our
solar system. We can use the GPS units outside to develop a “people-model” of a solar
system. We should first decide how big our model sun is. How big should (can?) we set
it so that we get a reasonable/doable model? We don’t want to be walking all the way to
the next state. After that decision is made we should get into several teams, with each
team having at least one GPS receiver. Then we’ll go outside and follow the steps listed
here:
1. Decide and Mark (both on the GPS and on the ground) the point we will designate as
our Sun. Each team should mark roughly the same point. I say roughly in that there is
measurement discrepancies due to the accuracy of the GPS units.
2. Each group will walk (in different directions) and mark (on the ground and the GPS)
each planet’s position in our scale model as they go along.
3. Pick out landmarks to describe as you reach each planet’s orbital distance.
4. Make note of the relative distances between the planet’s not only the distance from the
Sun.
Questions to consider upon returning to the classroom:
1. What do you notice about the distances of the inner planets (The Terrestial
Planets), from each other and from the Sun?
2. What about the outer planets (The Jovian Planets)?
3. If you wanted to place a scale size object at each of these points, what would you
think of using (this may be helpful with your students to visualize the immense
size differences)?
4. Additional science (Astronomy) extensions can be explored: What makes a
terrestrial planet different from a Jovian planet?
Another interesting relationship that can be explored with GPS and Astronomy.
Knowing how to find the North Star (Polaris) can help us determine our latitude.
Students should be encouraged to create a pseudo-sextant by taping a straw down the
length of an angle ruler as shown. This was an actual measure I made using my angle
ruler and then taking a picture and measuring it in Geometer’s Sketchpad.
Students should also be familiar with the coordinates that they currently reside at.
Example: Chambana is approximately 41degrees North. This can be found with the GPS
system.
Students should be encouraged to find the constellation URSA MINOR in the night sky.
With their makeshift sextants they should stand in a clear area after night fall and peer
through their straws at the star Polaris. They should then record the measurement on the
angle ruler. It may be helpful for them to sketch their findings. The following is an
example but more crude drawings are also acceptable. Students should be careful to keep
the bottom portion of their angle ruler parallel with the ground.
First, the location of Polaris and then the use of Sketchpad again to determine the angle
from my view. Ursa Minor will rotate about Polaris through the course of the night but it
always looks like the following type of figure (a scooplike figure).
Questions to consider:
1. How well does this approximation of your latitude match the latitude given by the
GPS unit?
2. What would the measure of Polaris be near the Equator?
3. How about near the North Pole?
4. How about near the South Pole?