Introduce the topic by asking the students if they had heard of the GPS (they most probably do). Ask them where they use GPS. Almost every modern mobile phone has a GPS receiver.
Tell them about the different GNSSs. At the end, let them summarise the content.
Ask the students if they can explain how measuring the time of a signal transmitted by the satellites provides the distance to it? Help them by bringing up the analogue with the thunderstorm. There is typically at least one student that knows about it. See the introduction for details.
Ask them how counting too fast/slow changes the result.
Tell them that with the satellites, they will have to measure the travel time of radio signals, a special kind of light, instead of the sound of the thunder. Ask them if they had an idea of how one could determine the signal travel time of a given satellite. The answer contains clocks to measure time.
After having established that those signals are transmitted by satellites, you may want to show them where and how many satellites there are. This can be done via free apps for smartphones like e.g. GPS Essentials. However, receiving GPS signals within buildings may not be easy.
ACTIVITY 1 (OPTIONAL): IDENTIFYING EUROPEAN COUNTRIES
Materials: - Worksheet 1 - Pencil
During the actual exercise, where the students locate their simulated position, they have to use a map of Europe. As an additional task, the teacher can ask them to identify European countries on a blank version of that map.
Hand out worksheet 1, which is the same as the map used in the next activity, but without the names of the countries. If available and necessary, hand out atlases.
As an additional optional exercise, let the students name the capitals of those countries.
In a culturally diverse classroom, the students may want to talk about the countries where they or their ancestors come from.
Figure 7: Topographical map of Europe displayed in Lambert azimuthal equal-area projection. This assures a constant scale throughout the entire map. The country borders are indicated with grey lines (Alexrk2, https://commons.wikimedia.org/wiki/File:Europe_relief_laea_location_map.jpg, ‘Europe relief laea location map’, https://creativecommons.org/licenses/by-sa/3.0/legalcode).
The results can be checked with the map handed out in the next activity.
ACTIVITY 2: LOST IN NO-MAN'S LAND
Materials: - Worksheet 2 - Compasses (drawing tool) - Pencil - Ruler (at least 20 cm) - Calculator
Hand out Worksheet 2 and repeat the story given below:
Imagine, you were abducted by aliens and taken on a ride across the solar system. On your return, you were dropped off somewhere in Europe. You have no idea where you are, but luckily you have your GPS receiver with you that should guide you to a place from where you can receive help or return home.
But … oh no … the receiver is broken. Instead of showing your location on Earth, it only displays the signal travel time of four receiving satellite signals. You will have to do it all by yourself.
Since the clock in the receiver is not perfect, the time on the display can be off from the true value. But you can deal with that later. With the map you find in your pocket and the working calculator function of the GPS receiver, you should be able to get along.
Figure 8: Topographical map of Europe displayed in Lambert azimuthal equal-area projection. This assures a constant scale throughout the entire map. On Worksheet 2, the scale is represented as follows: 300 km = 1.3 cm on the map. The names of the countries have been added (for credits see Figure 7).
(Note: The map provided is only a 2-D model of the real configuration, i.e. the altitude of the satellites above the ground is neglected.)
In the first step, the students use information on their worksheets to determine their approximate location. The signal travel times in the worksheet table (see Table 1) deviate contain an offset from the correct time, simulating a GPS receiver clock that is several milliseconds fast.
The time the signals take to reach to you is given in milliseconds (1000 ms = 1 s).
Radio signals are transmitted at the speed of light, so you only have to use the constant value of that speed (c = 299792.458 km/s) to convert the time into a distance.
Let us assume that the measured signal travel time is 10 ms (milliseconds). The measured distance to the satellite can be calculated as follows:
Δs = c • Δt
Δs: distance to the satellite
c: speed of light
Δt: signal travel time
Δs = 299792.458 km/s • 10 ms = 299792.458 km/s • 0.01 s = 2997.92 km
Instruct the students to always translate the time from milliseconds into seconds.
What is the scale of the map?
The first step is to determine the scale of the map. This should be done as accurately as possible. There is a line indicating 300 km. Important: Do not change the scale of the map in the worksheets. It indicates that 300 km on land correspond to 13 mm on the map.
How far away are the satellites?
Explain that the students should use the table on the worksheet (Table 1) by filling in the missing numbers. For this part of the activity, only the columns labelled ‘measured’ will be used. The columns labelled ‘corrected’ are only needed for the additional activity for advanced students (see below).
Table 1: Table as provided in Worksheet 2.
Now, the students are supposed to determine the distances the signals have travelled from the time displayed on the receiver (see table). Minding the units, the times have to be multiplied by the constant speed of light (299792.458 km/s). The calculated values are entered under the column ‘Distance (km), measured’.
Using the scale of the map, the students convert this distance to the one on the map. As mentioned before, 300 km on Earth correspond to 13 mm on the map. This value is then added under the column ‘Distance on map (cm), measured’. The resulting values are given in Table 2.
The students then use their compasses and draw circles or arcs with radii equal to the signal travel lengths. With each additional arc, the students are supposed to determine the countries they are in.
Question: What do you notice about your likely position with each additional satellite signal arc?
Answer: With each additional satellite arc, the position is determined with a higher precision.
After drawing the arc for the last satellite, the students will see that the four arcs do not intersect in a single point, but in an area that should include the true location (Figure 9). This is due to the problems in the receiver clock. The correct position is somewhere in the area enclosed by the intersections.
Question: What are the possible countries of your current position?
Answer: Possible solutions are the Netherlands, Belgium, Luxemburg and Germany.
Figure 9: Topographical map of Europe displayed in Lambert azimuthal equal-area projection. The arcs represent the distances that the signals have travelled from the four satellites in the times specified in the table of Worksheet 2 (for credits see Figure 7).
To determine the location more precisely, e.g. in order to be prepared for the extension activity below, a single point must be identified that represents the true location. Two approximations can be used for this.
The centre of mass of the resulting area of intersection can be estimated by guessing a point that has the same distance from all four intersections of pairs of arcs. It should be in the southern part of the Netherlands.
In real life, GPS receivers change the time offset common to all receiving signals until a point of intersection is found or the location is narrowed down to an small area. This algorithm can be simplified in this example by approximating the centre of mass graphically. The common area is surrounded by four arcs. The students have to determine the bisectors for each of them. Then, they connect the opposing bisectors with lines. This results in two crossing lines whose intersection can be defined as the centre of mass (Figure 6). It is a good approximation of the true location. Again, the location should be in the southern part of the Netherlands (Figure 10).
Figure 10: The same map as before, but this time with the approximation for the simulated location (for credits see Figure 7).
EXTENSION FOR ADVANCED STUDENTS: CORRECTING THE CLOCK
Question: What is the reason for the mismatch between the distances derived from the GPS measurements and the location determined by interpolation?
Answer: The clock of the GPS receiver is incorrect.
In a real GPS receiver, this interpolated position is used to calculate a correction for the satellites. This is then assumed for subsequent positioning. This additional activity illustrates these calculations and determines the offset of the receiver clock.
After determining the centre of mass of the area of intersection (Figure 10), the students measure its distance to each of the satellites on the map. The values are added to the table on their worksheets in the column labelled ‘Distance on map (cm), corrected’. These numbers are then converted into true distances using the map scale and added to the table in column ‘Distance (km), corrected’.
The corrected signal travel times for each of the satellites are calculated using the speed of light and added to column ‘Signal travel time (ms), corrected’. With this, the students calculate the difference between the initial value given by the simulated GPS receiver and the corrected one. These differences are added to the column ‘Correction (ms), meas-corr’. From the resulting four differences, the students derive the average. This is the clock correction the simulated GPS receiver would have to apply to any additional measurements. The resulting numbers would be similar to the ones in Table 2.
Table 2: Table with results.
Note that due to uncertainties introduced by the drawing on the map, the numbers can vary a bit.