"A PROJECT FOR ELOS INTERSCIENCE"
Gert Kiers
RSG Tromp Meesters Steenwijk
The
Supervision
Jan de Jager
RSG Tromp Meesters Steenwijk
The
Introduction
Erathostenes' size of the Earth and Measuring the Distance to the Moon
Schoollife becomes definitely more interesting if there is an international
touch to learning. Experiments like determining the size of the Earth or
measuring the distance to the Moon need this international touch. Only if you
have at least two locations far apart, say approximately
The goal of these partly hands-on, partly theoretical exercises is to confront pupils with the principal difference between mathematical solutions to physical problems and physical measurements. Mathematics is (in secondary schools) exact and precise, while physical experiments are always subject to experimental errors. These errors usually "destroy" your expectations you got from the mathematical analysis.
Both the Erathostenes' experiment and the Distance to the Moon need careful preparation to determine the exact times the height of the Sun or Moon should be measured. This requires some communication skills between two or more schools. Stress can be expected here.
The Erathostenes'
experiment is similar to the one the Greek geographer used during the great
Greek civilization. Instead of using a well in Syene
and a pole in
Measuring the distance to
the Moon with very simple instruments is very difficult, because the accuracy
of the measurements is vital. We used a long pole that cast a shadow of the
(near) full Moon. In another case we used a
Unfortunately, this experiment is bound to result in distances which are rather far off the real value. The difference of the angles measured at two distant locations, is very small (typically 0.5 degree). However, obtaining a deviation can also be very educational. It shows pupils that mathematical theory and physical practice are two separate worlds, both of which need to be treated with respect.
We would very much like to repeat these exercises to work towards more precise values by thinking about better equipment.
The project involves in lessons
Reference: math method Getal & Ruimte, NGNT 3, exercise 63
Assignment: How to calculate the distance to the moon?
In order to calculate the distance from
One chooses a place at the same meridian as
At the same moment in
The measured angles area in the figure marked with a and b: a = 37,72 and b=11,03.
Because
g = 48.
Besides use the radius of the earth is
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a. Calculate AB in the triangle ABM.
b. Prove that BAC = 76,28 and calculate ABC and ACB.
c.Calculate the distance from
The project calculating the distance to the Moon
We will try to calculate the distance to the Moon by using trigonometry (that is sine, cosine, tangent). Fig. 1 shows the situation if you look at it from a mathematical point of view. We go into detail later on. The method is very straightforward and very easy if you know how to use sinus, cosine and tangents.
1. The method
You observe the Moon from two cities (well, actually somebody else
does this in that other city) at approximately the same local time (for
instance, at 8 o'clock at night on 25 December). In both cities the angle
between the horizontal (= direction of the surface of the Earth, the horizon)
and the direction of the Moon (see Fig.2) must be measured.
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Fig. 2.Which angle do you measure?
If you then know where on Earth the other city is, you can calculate all angles and sides of a triangle (see Fig. 1) using the sine-rule (also known under other names like 'law of sinus' or 'rule of sinus').
2. The mathematics
In Fig.1 you find the angles involved in this problem. Let us first have a look at the angles at the observation points (= where you stand). The method to calculate the Moon distance needs the distance between cities in a North-South direction only (that is, in a straight line through the Earth; not alongits curved surface). Therefore we need only the latitude coordinates of the cities, and not the longitudes. This method requires that observations need to be done at the same local time (e.g. 8 o'clock), so that the Moon is in approximately the same position in the sky. This requires some brain teasing to understand, but it is fun to play with. It helps if you take a globe of the Earth. You can also try to figure out that if we use the real distance between cities by using the longitude coordinates as well, we need to observe at the same universal time. However, this requires complex formulas.
Let us first look at the situation in the northerly lying city.
a. The northern city
See
Fig. 3. We measure angle a1
(we find this out later).In Fig. 1 we see that this is not the angle we look for. There is an angle a2 that needs to be added. How do we calculate this angle
a2? Look at triangle ABC in Fig. 1 and in close-up in
Fig. 3. Side AC (= radius of the circle) is perpendicular to the horizontal
in A (that is so in any circle).
Thus, a2+a3 = 90°.
(Well, instead of knowing a2
by now, we have another angle we do not know. Where is the progress here!).
Now, how do we find a3,
then? For this we look in triangle ABC. Angle g is the difference between the latitudes of the two cities (see Fig.
1). As we know the radius of the Earth,
And now the southerly lying city.
b. The southern city
It is nearly the same as for the northern city. If you look in Fig. 4 you will see that the angle b in Fig. 1 is now 180 - b1+ b2. The angle b1 will be measured, and angle b2 is equal to a2; both are calculated in exactly the same way (see northern city). So now we know b in Fig. 1.
c. The distance between the cities
From Fig. 1 we see that the distance m (= length of AB) can be calculated easily by using the sine-rule in triangle ABC.The radius r is known (6,380103 km). The angle g can be calculated from the difference in latitudes of the cities (or, if one has a GPS system, even the latitudes of the observing site).
d. The distance to the Moon
So, we now know angles a and b. We also know the size of side m. Therefore we can determine sides a and b by using the sine-rule, and thus we have the distance to the Moon. Easy, isn't it? You can calculate the distance to an object that is very far away by simply measuring two angles.
But how do we measure the angles?
3. The physics
Measuring things requires physicists and engineers. One can measure angles in a variety of ways. We will talk only of those that lie within our possibilities to actually perform. Nothing like laser ranging, radar techniques, sending space probes, and the like.
The main problem with the distance to the Moon is that this Moon is very far away. You will find that the angles a and b in Fig. 1 are therefore almost equal. Subtracting or dividing two numbers that are almost equal calls for big trouble. The only things to prevent "disaster" are either to come up with a totally different method of measuring, or to measure very accurately.
For this project it is fingers crossed, but you will see how this problem works, and this may even save you from blundering in your later career.
The instruments we can use
a. Theodolite
The most sophisticated instrument we hope to be able to use is the theodolite. It is a small telescope which land surveyors use to measure distances between houses, streets, cities, etc., or to determine heights of land or buildings, or to lay out courses for roads or rail tracks. Basically you measure angles with theodolites. It can be done very accurately. How to use a theodolite will be instructed if there is one available.
b. Gnomon
A gnomon is nothing else than a stick or a pole. You put it straight up and look at its shadow as is cast by the (full) Moon. Measure the length of the shadow and the length of the pole as accurately as possible. You will then be able to calculate the angle a1 or b1 using the inverse tangent. See Fig. 5.
Fig. 5. The angle measured with a pole
Regrettably, the gnomon is an imprecise instrument. If you calculate how accurately you need to measure the pole and its shadow, then you would see that the measurement should be accurate to approximately a millimeter if you have a pole of three meters high. The length of the shadow is thus very difficult to measure sufficiently accurate.
Other methods - Observing in Daylight
As it is not always attractive to observe at night, it
is also possible to observe during school hours or shortly afterwards. The
instrument we will use (with some varieties) is as follows. The figure shows a
long (
Instead of using a hollow pipe, one can use any kind of vizer type instrument, e.g. put two rings far apart on a long pole. Looking through the ring in the direction of the moon gives a the angle under which you see the Moon. Measuring some sizes in the triangle should give you the right angle.
Another method would be to use one fixed point (e.g. a well visible pin on a roof, while the observer moves on the ground until the Moon and the pin are lined up. Then, one still needs to determine the position of observer accurately, but it should be feasible.
Again, the great advantage with such instruments is that you can work in daylight. This is advantageous if your school is difficult to use in the evenings. Closed down, far away, etc. With this method you can work within the school hours.
But again here, the method is not highly accurate.
Two other ways of measuring the angle of the Moon's position above the horizon. Above: a long hollow pipe; below: two rings or pins far apart on sticks.
Solutions
But, there is no problem without a solution. Again mathematics helps (may help) here. There are at least two ways to improve the accuracy of the result of the measurement.
1/ Statistics
Do the measurement with many people. Each person measures the length of the shadow as accurately as possible. Then you have a number of measurements from which you calculate the average. The more people, the better the result. The measurements should be independent. So, do not mention your result after you have measured it, but write it down on a piece of paper. Otherwise others will hear the number, and this will influence their measurement.
2.Fitting a curve.
Do your measurements of the shadow in the course of two hours. One hour before the moon goes through the meridian (that is, is it stands exactly in the South), and one hour after this. Measure the length of the shadow every four, five minutes or so. Put your measurements in a graph. This will look like Fig. 6. You will need a small programme (usually a good spreadsheet has one) that calculates the parameters of the line that fits nicely with these points. This is called curve fitting. From this you will be able to calculate the length of the shadow at a certain time much more accurately than with a ruler or a measuring tape.
Now, you need to be careful with choosing this curve-fitting method and the phase of the Moon (first quarter, full moon, etc.). If you choose Full Moon or close to it, you will be observing in the middle of the night! I have no problem with that, I do not know about others.
One or two days after First Quarter would be nice. The Moon then passes the meridian at about seven or eight o'clock. The light should be bright enough to cast a shadow in the Winter.
Fig. 6. Measuring shadows in the course of two hours
4. The astronomy
After finding the angles and communicating the results with other groups you can calculate the distance to the Moon. Accurately? Maybe, but it would be nice if you had fun.
5. The organization
It is clear that some coordination should be done in order to have at least two groups working at the same time. Each group should appoint a coordinator (and a vice-coordinator, in case of absence of the coordinator) to keep in touch with the other group coordinators and their local teacher.
The coordinators should exchange information regarding:
1.time of observation (watch the weather)
1a. a large crater on the Moon to point the instruments at for improved accuracy. This may be useful if each group uses a theodolite.
2. results from other groups (email or through internet conferencing)
3. final results
4. comments on other observations and suggestions for improvements
It is by all means possible that a group has a different coordinator for each task (is also more fun).
Observations need not be done for one night. If there is another opportunity later, you can coordinate a new observation. For example, if a group has obtained a theodolite or have thought of a more accurate method, or if the first measurement was totally wrong.
This experiment is set up from a thought. Science and technology work in the same way. The experiment may work, it may not work. You never know. Although this experiment will not be a scientific breakthrough, nor will be a Nobel prize awarded for it, it does have a few elements that you will also find in scientific experiments: curiosity, fiddling with numbers, hope, despair, and fun, as science often is.
I have not found a (hi)story that people have done it in this way. I now know why not. You find out for yourself.
Fig. 3. Angles for the northern city A
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Fig. 4. Angles for the southern city B
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