Astronomy Online

Determining the Earth-Moon Distance

1. A brief introduction

This Astronomy On-Line Collaborative Project - finding the distance to the Moon - is probably the first of its kind.

You have probably already had a look at the simplified 2-dimensional (2-D) method.

The following pages provide details about the 3-dimensional (3-D) method used to determine the Earth-Moon distance by means of a solar eclipse.

The European Student Project Group wants to find the optimum method, in order to prepare in the best possible way the observaitons which will be made at the time of the Total Solar Eclipse of 1999 visible in Europe.

So - below we present a more elaborate 3-D method too. We hope that many of the groups that participate in the Astronomy On-Line Programme will take this opportunity to try it out.

The 3-D method takes a bit more of work, but has the advantage that you do not need to know the accurate values of the lunar and solar declinations as you do for the 2-D method.

The present project takes advantage of the solar eclipse on October 12, 1996, to compare the relative positions of the Moon's and the Sun's centres as observed from two different locations at exactly the same time (UT).

2. The basics of this method

Hold a pencil at arm's length in front of you and alternately close your left eye, then your right eye, aiming your sight at a distant landmark (for example, a church spire). When looking with your right eye, the image of the pencil seems to have shifted to the left of the landmark, whereas when looking with your left eye, it seems to have shifted to the right of the landmark (Figure 1).



Figure 1.

2.1. A Model

To a first approximation, our eyes can be compared to two `camera obscuras' (Black Boxes with a pinhole in the front through which the light passes) whose screens show an inverted image of the landscape, just as the retinas of our two eyes do.



Figure 2.

Our brain processes what our eyes see and, so to say, `inverts' the final image. It is therefore quite possible to replace the two "black boxes" by a single box and get the final image in the correct position (Figures 2 and 3).



Figure 3.

It can be seen on Figure 2 that T1CT2 = C1TC2 = a. Note that the angle a is indicated as `alpha' in Figure 2.

As a is small, then D/L ~ a.

On the screen of the Black Box, the distances C1 and C2 between the two images of the pencil correspond to angle a.

This angle can thus be associated to the measurement of a length: one has just to know the distance corresponding to the unit angle, i.e. 1 radian (rad).

D is the distance between the two parallels, starting from both eyes and pointing at the landmark. Knowing D, we may derive L from L = D / a rad (see Figure 2).

2.2. What is the relationship with an eclipse?

Replace the top of the distant landmark by the centre O of the Sun, the horizon line by the apparent diurnal path of the Sun, the tip of the pencil by the centre C of the Moon, then imagine one Black Box at Stockholm (Sweden) and another at Toulouse (France), both being oriented so that the image of the Sun is at the centre of the screens.



Figure 4.

Then watch the motion of the Sun for 4 minutes. On the day of the eclipse (October 12), the apparent path of the Sun is fairly close to the celestial equator (the Sun's declination is about -7 degrees).

Since the Sun moves a full circle (360 degrees) in 24 hours, the arc travelled by the Sun's image in 4 minutes is equal to 1 degree.



Figure 5.

3. Activities

3.1. Required equipment and tools

- cardboard plates to blind the window;

- 3 small `Sun' discs (adhesive material) with a small hole carefully drilled in their centres;

- 1 small `Moon' disc (adhesive material) with a small hole carefully drilled in the centre;

- a cardboard or plywood plate and any relevant gadget to keep it at an angle with the floor;

- 1 sheet of white paper to be used as a screen;

- adhesive tape; and

- two T-squares

4.2. Preparing for the measurements

- Select a window looking to the South;

- Blind it with cardboard plates;

- Drill a small hole the diameter of a nail so as to collect a thin beam of light from the Sun;

- On the floor, as far as possible from the window, set the cardboard or plywood plate with the sheet of paper pasted on it, at a convenient angle with the floor; and

- Using the T-squares, incline the plate so that the beam of light from the Sun should fall at a right 90 deg angle on the screen then keep the device firmly in place.



Figure 6.

You must do it one or two days before the eclipse, and you must cut in advance the solar and lunar disks at the right size.

3.3. How to do the measurements

- select any time t for the experiment (such as 14:00 , 14:15, 14:30...);

- stick a first `Sun' disk S1 on the image of the Sun at epoch t - 4 min;- proceed likewise with a second `Sun' disc S2 at epoch t;

- at the same epoch, stick the `Moon' disc as shown on Figure 7; and

- finally stick the third `Sun' disc S3 at epoch t + 4 min.



Figure 7.

3.4. Using these observations

- check that the centres of the "Sun" discs are well aligned: the straight line thus obtained will define an axis. This will be referred to as the abscissae axis positive toward S3.

As 4 minutes of time correspond to 1 degree, the axis can be graduated in degrees and its sub-divisions (arcminutes).

- from the centre of disc S2, draw a straight line perpendicular to the abscissa axis and graduate it using the same unit as on the first axis. This will henceforth be referrred to as the ordinate axis (positive toward the same side as the center of the Moon).



Figure 8.

- in the reference frame thus defined, measure as accurately as possible the coordinates (abscissa and ordinate) of the centre of the `Moon' disk.

- If possible, it would be interesting to make measurements at several times: this will make it easier to collaborate with other groups.

4. Communicating, Exchanging and Processing the Data

4.1. What must be sent over the network

The information will be circulated on the Astronomy On-Line Web-sites. The requested data are:

- longitude of the site;

- latitude of the site;

- epoch t in Universal Time (ex: UT 14:00, 14:15..);

- coordinates (abscissa and ordinate) of the centre of the `Moon' disk

4.2. Determining the Earth-Moon distance

As soon as they are available, get from the Astronomy On-Line Web-sites the results sent by two different `observatories' (groups) that should be as remote as possible from each other and should have made their measurements at the same epoch.

- plot the results on a diagram as shown on Figure 9;

- measure the C1C2 distance;



Figure 9: Determining the parallactic arc C1C2

- convert the measurement into radians;

- determine L from L = D/a rad

The distance D is determined with the help of software that may be downloaded from the Astronomy On-Line Software Shop. The requested data are the geographical coordinates of the observing sites and the epoch of observations (UT).

4.3. Share your comments!

The e-mail service is meant to collect your impressions, criticisms, suggestions and your results.

Thank you for sending us your comments!

Please send your reports, comments, etc. by email to the European Student Project Group.

We will keep you all informed about the progress by means of short status reports which will appear in the Astronomy On-Line Newspaper.

Have a nice hunt and enjoy your eclipse!

Contacts

From here you may contact the members of the European Student Project Group :

Josée Sert

Francis Berthomieu

Brian Stockwell

Anders Västerberg

Mogens Winther

And here is the European Student Group, snail mail Address List,

Credits

The drawings from this page may be reproduced, in case Astronomy On Line and the EAAE are mentioned.


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