## Students measure the Lunar Distance...## 2-Dimensional EAAE-method
## A brief introductionThe following pages provide an introduction to the 2-dimensional (2-D) mathematics explaining solar eclipses.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 make an extremely accurate determination of both the distance to the Moon as well as its size
You do not have to read all of this mathematical in order understand
and to participate in this On the other hand, the mathematics here are ## Seasonal VariationsYou have (probably) all noticed that and So, the `height' of the Sun (above the horizon) varies from season
to season. This height is referred to as the 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 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 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. These topics were discussed in details during the 1989 EAAE-ESA-ESO Sea and Space Project Navigational Chapters. Daily values for the Sun's declination may be found in the Sea and Space Almanacs, Books for Scouts, etc. ## Lunar and solar declinationDuring 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 some of the hourly values for one particular day, August 11, 1999, as they may be found in naval handbooks, or as given by NASA Let us take a look at the declination values, as they appear on
this date, at 10 hours 23 min UT = 11 hours 23 min British Summer Time,
or 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 15.8920 degrees. The Sun will have a declination a bit below that: 15.3359 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 15.89 - 15.34 = 0.55 degree, that is, they are more than half a degree apart. This is more than the sum of the radii!
## The parallax effectHowever, don't worry, the eclipse on August 11, 1999 In order to explain what may seem like an obvious discrepancy, we need to discuss some trigonometry. From our daily life, we are familiar with the geometrical effect
that is called 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 The same type of geometry is apparent during the solar eclipse on
August 11. When viewed from the northern part of our globe, the
comparatively (in astronomical terms!) Now imagine two observers, one at position 1 and the other at position 2, which is further to the North. In general - observers have a globe position defined by their Longitudes and Latitudes . You may find your own geographical latitude in any geographical atlas, or by means of this WWW interactive map server. In our 2 Dimensional Model, we concentrate on the geographical latitude. Observer 1 has a geographical latitude relative to Equator, marked with yellow colour. Now let us get back to the solar eclipse. From the drawing above, it is obvious that observer 1 sees the Moon at a lower angle than does observer 2. So, if we move from position 1 towards more northern latitudes, we will
experience that the lunar declination appears to 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
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 half a degree apart from each other in the sky. ## Trigonometry of the eclipseThe 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
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
The geographical latitude of London is +51.5 degrees. The Lunar declination is 15.89 Deg, and the radius of the Earth is 6378 km. We therefore only need to estimate the parallax anglealpha
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 15.89 - 15.34 = 0.55 degree. 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): Please notice, not only has the previous 0.55 Degree distance vanished, the Sun and Moon even appear to have interchanged position. As seen from London the Moon is thus shifted with 0.55 Degree + 0.03 Degree equal to 0.58 Degrees. Now try this exercise: This result is
Click for additional Examples, based on the October 1996 Eclipse However, at this point a few comments must be made: *Real life will be different, the values above were taken from a "perfect computer sky program".**We only apply simple, inexpensive tools.**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 Central European Countries.*
However, during the partial eclipse 1996, EAAE contributors Europe Wide yielded a quite precise measurement of the lunar distance. Click for the comprehensive 1996 report Due to improved geometry (close to the "meridian" - south) we may during the August 1999 eclipse expect even better results. ## What is the size of the Moon?During the August 11 eclipse, the Moon's angular radius will be approximately 0.27 degrees. You will be able to measure this angular radius yourself, for instance by means of the Pinhole 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, expressed in kilometres. ## ConclusionsBy 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!
Once you have made your observations, please do not forget to send
your Please send your reports by email to the Have a nice hunt, remember General Gordon and enjoy your eclipse!
The drawings from this chapter may be reproduced, in case Astronomy On Line and the EAAE are mentioned. |