Astronomy Online

Mathematics of Solar Eclipses (2-D method)

A brief introduction

The following pages provide an introduction to the 2-dimensional (2-D) mathematics explaining solar eclipses. A 3-D Method is also available.

This method serves to explain the geometry of solar eclipses, by considering the Moon's position at the time of an eclipse. Provided many observations are available, it may become possible to determine the distance to the Moon as well as its size from thses calculations.

You do not have to read all of this and to understand the details in order to participate in the Astronomy On-Line Collaborative Project which is concerned with the Solar Eclipse on October 12, 1996. But we strongly recommend that you try to obtain the observations, as described in the main document. When taken together with the observations made by other groups, they will contribute to these calculations.

On the other hand, the mathematics here are not very complex and should be quite comprehensible to secondary shool students who have studied basic trigonometry.

Seasonal Variations

You have (probably) all noticed that during winter time the shadows are long (right):

and during summer time the shadows are short (left).

So, the `height' of the Sun (above the horizon) varies from season to season. This height is referred to as the altitude by astronomers.

Astronomers usually describe this variation by considering the conditions at the Earth's Equator:

Note that the solar rays hit our planet at a certain angle with respect to the Equator. Astronomers call this angle the solar declination.

In the situation above, the Sun is positioned so that we experience Summer in the Northern hemisphere. Mathematically speaking, at this time our Sun has a positive declination.

The solar declination reaches a maximum at +23.44 degrees around June 22.

During wintertime in the Northern hemisphere, the Sun reaches its minimum declination, -23.44 degrees, around December 22. Halfway between these two dates, that is, around March 22 and September 22, the Sun's declination is near 0 deg. The daily values for the Sun's and the Moon's declination may be found in Civil Almanacs, Books for Scouts, etc.

Lunar and solar declination

During many centuries, precise lunar and solar declination values have been of crucial importance to sailors, navigating on the seas by means of instruments like the sextant. Here are the some of the hourly values for one particular day, October 12, 1996, as they may be found in naval handbooks, like the Nautical Almanac, or the Astronomical Almanac 1996 (on page A80).

Let us take a look at the declination values, as they appear on this date, at 14 hours 24 min UT, or 16 hours 24 min Central European Summer Time (CEST). This is the time the solar eclipse on that day reaches its maximum, as seen by an observer in the French capital, Paris. That is the moment when the Moon covers the largest fraction of the solar disc.

From the Almanac values above, we find (by interpolation or by plotting the table values in a graph) that at this time, the Moon will have a declination that is equal to -06.69 degrees. The Sun will have a declination a bit below that: -07.64 degrees.

Both the Moon and the Sun have a diameter (as seen in the sky) that is close to 0.5 degree. The radii of the Sun and the Moon are indicated in the figure below:

The centres of the Moon and the Sun are thus spaced in the sky by 07.64 - 06.69 = 0.95 degree - that is, they are approximately 1 degree apart. This is more than the sum of the radii! So, at a first glance, it seems that the Moon is too far away from the Sun in the sky to yield any eclipse at all!

The parallax effect

However, don't worry, the eclipse on October 12 WILL happen. Here is what people in Paris will actually observe:

In order to explain what may seem like a discrepancy, we need to discuss some trigonometry.

From our daily life, we are familiar with the geometrical effect that is called stereoscopy. In astronomy this effect is known as parallax.

If an object is at close distance, your left eye and your right eye will give two different views. Nature knows this effect, in fact this is how we are able to estimate distances. If you hold a pencil at a short distance in front of your eyes, you may observe an effect like this:

Note that if you move the close-by pencil out to large distances, this parallax-effect will become smaller and smaller and at some time it will approach zero and vanish. The church is too far away to give any noticeable parallax.

The same type of geometry is apparent during the solar eclipse on October 12. When viewed from the northern part of our globe, the comparatively (in astronomical terms!) close-by Moon is observed in a slightly different perspective.

Now imagine two observers, one at position 1 - say in Paris, and the other at position 2, which is further to the North.

It is obvious that observer 1 sees the Moon at a lower angle than does observer 2.

So, if we move from Paris towards more northern latitudes, we will experience that the lunar declination appears to decrease. You have probably been in a similar situation: As you climb a tree, you will `look more and more down' on all your neighbours. This is also the parallax effect.

EAAE schools have tried to apply this principle once before. In March 1996, Comet Hyakutake passed close to the Earth. EAAE students in several countries attempted to measure the distance to the comet by means of a North Europe - South Europe parallax effect. They were quite successful.

You may perhaps know that astronomers frequently apply this geometrical effect by combining pictures of star fields taken half a year apart, when the Earth is at opposite parts of its orbit around the Sun. On such pictures, the images of relatively nearby stars are seen to be shifted with respect to the background stars which are much farther away. From such observations, it has been possible to estimate distances to many nearby stars.

It is thus because of the parallax effect that this partial solar eclipse will be visible at all. Without this parallax effect, the Moon and the Sun would still be nearly one degree apart from each other in the sky.

Trigonometry of the eclipse

The next drawing shows in more detail by how much the lunar declination is decreasing, when we move from position 2 to position 1:

Note, how the small angle - alpha - (indicated by the greek letter in the figure) is present in two places. The drawing below shows how we may find alpha:

The shaded triangle below will be essential for all our subsequent calculations:

Any High School Mathematical text book will tell us about the famous sine relation:

If we apply this formula on the shaded triangle, we find:

The distance to the Moon

Let us now apply the last formula to the Paris example. Note that we know all values:

The geographical latitude of Paris is +48.9 degrees. The Lunar declination is -06.69 Deg, and the radius of the Earth is 6378 km.

We therefore only need to estimate the parallax angle alpha in order to know all sides and angles of this very long triangle.

At the beginning of these notes, we found that the Sun-Moon distance, as seen in the sky, `ought to be' equal to 07.64 - 06.69 = 0.95 deg. However, due to the parallax effect discussed above, the Moon was `artificially lowered' in the sky, so instead we get a picture like this one (the data has been taken from a planetary software program):

Thus, the observed distance in the sky between the geometrical centres of the Moon and the Sun is only 0.20 degrees. Put in another way, when seen from Paris, the Moon is shifted with a parallax equal to alpha = 0.95 Deg - 0.20 Deg = 0.75 Deg.

Now try this exercise: Use your pocket calculator. Show that the formula above, and the data here given implies that the distance to the Moon, at the time of the eclipse, is approximately 400,000 km.

From advanced astronomical tables, it is known that during the solar eclipse on October 12, the distance to the Moon will be approximately 384,005 km.

So this method seems to be quite promising!

However, at this point a few comments must be made:

1. Some of the values used here must first be observed. In the above example, the Moon-Sun sky distance of 0.2 degrees, at the time of the maximum of the eclipse, was not an observed value, but had been calculated in a sky simulation programme. As described in Sky & Telescope (September 1996), some of the programmes which are available for amateurs and schools do not produce very reliable results. Thus, we have to wait for the real observations which will be carried out by the Astronomy On-Line groups. Then we will see whether the results we get will be even better?

2. We only apply simple, inexpensive tools.

3. So far, we have only described a simple 2-D method. This method works best when the Sun is close to the meridian, that is when it is seen due South. This may slightly favour observers in countries like Iceland, Portugal, Great Britain, Spain and France. The more advanced 3-D method avoids this problem, but it requires more careful preparations.

What is the size of the Moon?

During the Octoner 12 eclipse, the Moon's angular radius will be approximately 0.26 degrees. You will be able to measure this angular radius yourself, for instance by means of the Black Box Camera measurements.

Once you have measured this angular size, and you have calculated the distance of the Moon, you may also determine the Moons real diameter, xpressed in kilometres.

Conclusions

By taking the average of several results, we may in joint cooperation perform the first reliable, amateur/student based estimate of both the distance of the Moon and its diameter. The more observations become available from observers at different locations, the better will be the accuracy!

So do join in!

Once you have made your observations, please do not forget to send your report back to us, as soon as possible.

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

Have a nice hunt, remember General Gordon 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 Project Group, snail mail Adress list,

Credits

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


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