Measure the distance from Earth to the Moon using high-school geometry and an international network of schools and observatories.
Imagine stretching your arm out and looking at your thumb, first with one eye, then with the other. The apparent shift of your thumb with respect to the background is called parallax. The same principle applies if two different schools ‘look’ at the Moon: they will see it slightly shifted with respect to the stars in the background.
In this activity, schools in different continents pair up so students aged 16-19 can compare their observation of the Moon across distances and calculate Earth’s distance to it (figure 1). Equipped with only a good camera and a good knowledge of geometry, the observation takes approximately 2 hours and the calculation 3 hours (establishing the partnership might take a little longer…).
The overall observation plan for the activity is detailed in figure 1, where M is the Moon, depicted as a point because its size is very small compared to the distance calculated (approximately 1/100th).
As in many scientific endeavours, planning is key. In this case, in addition to defining the right conditions to make the observations, the teacher needs to determine what margin of error is acceptable: this is important so the pupils are not disappointed if they don’t find the exact distance. Below we list a number of important points to highlight when discussing the activity with the class, but also when setting up the partnership with another school.
Figure 1 shows how two observers (A and B) will see the Moon, M, as being in two slightly different positions in the sky. While in practice the two observation points, A and B, M, and the centre of Earth, C, do not lie on the same plane, to simplify the calculations so that we can use plane trigonometry alone we assume that they do.
For that approximation to be as exact as possible, the two observation points need to be at the same longitude, and the Moon should be at its highest point (in culmination)w1 at the moment of the observation. This ideal situation is very difficult to obtain but we recommend that you stay as close to it as possible, and that you are aware of the errors implied by sizeable deviations from such conditions.
Moreover, if the Moon’s angle with the celestial equator (declination)w1 is equal to the average of the latitudes of A and B, ABM forms an isosceles triangle and this further simplifies the calculations.
You will also need at least two bright starsw1 (or planets) in the background to find the two apparent positions of the Moon, MA and MB.
Then we will consider that the two straight lines AMA and BMA are almost parallel, as are AMB and BMB. This would mean that the angles α ≈ α’ (figure 1). While not strictly true, this assumption is acceptable, as the pairs of lines converge far from both Earth and the Moon. Of course, this appears to be very far from true in figure 1, because it is not drawn to scale.
If we measure the angle α and the distance AB – known as the baseline – plus another angle in the triangle ABM, we can calculate all the other distances. Otherwise, we can make the triangle ABM become isosceles, and knowledge of a and AB is sufficient to calculate all the distances.
A key point is that the baseline must be long enough, when compared to the distance we want to find, to prevent the parallax from becoming vanishingly small. For the Moon, a distance between the partner schools of around 1000 km is enough, but the larger it is, the better.
Despite all the care that will go into choosing the best conditions, the measurements will not be perfect. The main sources of error are:
Errors due to d) and e) are not very important: our measurement is based upon large angles and so is not affected greatly by an imprecision of a few arcseconds.
Time synchronisation (f) is not very important either because the Moon travels the equivalent of its own diameter in one hour, so an imperfect synchronisation of a few seconds (or even minutes) is not relevant.
Distortions due to camera lenses (c) can be reduced if a small angle of view is used, like the one provided by a telephoto lens. A normal camera lens introduces a larger, but still acceptable, error. In our case, that angle was not so small and we estimate it generated an imprecision of approximately 1-2%.
Errors due to a) and b) are the most important and can account for an imprecision of 5-10% each. Together, they account for an overall error of about 10-20%. To reduce a) we must choose a long parallax baseline so that the shift of the Moon is as large as possible; to reduce b) we must properly choose places and moments for observations so that A, B, C and M lie in the same plane. If both conditions hold (for us, the first condition did but the second didn’t), the error can be reduced to a few percent.
We set up a network of schools, observatories and educators across the planet to undertake this measurement. It is made up of the following members:
With this network, or their own, teachers can propose dates to carry out lunar observations to work out the distance from Earth to the Moon.
The long distances between the schools in our network provide a sufficiently long baseline (distance AB) to make it possible to measure Earth’s distance from Mars in May 2016 (Cenadelli et al, 2009; Penselin et al, 2014). At that time, Earth will be situated between the Sun and Mars, and Mars will be almost at its closest possible distance to Earth, an ideal position for such observations.
If you would like to contact any part of this international network to perform measurements, please contact Davide Cenadelli at email@example.com
The only specific material necessary is a good camera to take photos of the Moon and the sky. A telephoto lens with a focal length of around 100–200 mm is the best choice, but a normal lens will also work if the bright stars or planets in the background are not very close to the Moon.
If we drop the assumption that AMB is an isosceles triangle, we need to know the value of another angle, such as BAM. BAM is equal to the sum of β, i.e. the altitude of the Moon above the horizon from A, and γ (see figure 3). β can be measured with proper equipment or, in its absence, can be taken to be almost equal to the altitudew1 of one of the reference stars or planets we used to measure the shift of the Moon. For Procyon, we had β = 39.3°.
We can calculate γ using the value for δ that we calculated earlier and the geometry rule that says: γ = δ/2.
It follows that BAM = β + γ = 79.5°.
Finally, if we apply the law of sines to triangle ANM, we have:
ΝM/sin79.5° = AN/sin(α/2)
ΝM = AN (sin79.5°/sin0.6°)
≈ 385 536 km (4)
CM = CN + NM
= CAcosα/2 + NM
= 4866 + 385 536 km = 390 400 km (5)
This result is still realistic and only 1.9% less than the known value.
The authors wish to warmly thank all participants in our network, as well as the students who participated in the ESO Camp 2014 who performed a similar measurement.
The Astronomical Observatory of the Autonomous Region of the Aosta Valley is supported by the Regional Government of the Aosta Valley, the Town Municipality of Nus and the Mont Emilius Community. Andrea Bernagozzi has carried out part of the work for this project while supported by a grant from the European Union-European Social Fund, the Autonomous Region of the Aosta Valley and the Italian Ministry of Labour and Social Policy.