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II.4 Mathematical Exercise related to the Solar Eclipse on October 12, 1996

One of the Webpages under the Collaborative Project on the Solar Eclipse on October 12, 1996 contains a description of mathematical concepts of such an eclipse and why these eclipses occur.

Let us briefly mention a few of these arguments.

Here you see a schematic drawing of the Sun and the Moon :

Figure 1

This drawing is of course not drawn to correct scale - but it indicates two important angles, the solar declination, and the lunar declination.

The next table shows the values of both solar and lunar declination during the eclipse hours:

Figure 2

Please observe, there is actually a large difference between these two declinations (last two columns).

Math Exercise

In the second last row you find the motions per hour.

Calculate the motions per minute.

Now calculate the lunar and the solar declination at 14 h 23.5 min UT - max eclipse time as it was measured by the Orion Group in Ystad, Sweden. Compare with the results below - or the output from our 2D software.

This difference is described on the figure below - showing the lunar and solar declinations at 14 h 23.5 min UT.

Figure 3

So, at a first glance, it seems that the Moon is seen far too high above the Sun to yield any eclipse at all !

However, as you all know, the eclipse did happen !

Figure 4: Photo by Astronomy On-Line group SCIVIAS in Bochum, Germany. Click on image to see larger version (JPEG, 19k)

In order to explain what seems a discrepancy - we need to discuss trigonometry.

As described in our math-2D chapter - we here experience a kind of stereoscopic effect.

Observer 1 and Observer 2 won't see the Moon at the same position of the sky.

Figure 5

Actually, the moon is shifted by the angle alfa - see Figure 6:

Figure 6

Astronomers usually call this angle alfa for parallax- you will find more details in the Math-2D section.

Below we will repeat the simple relation between this angle alfa - and the distance from us to the Moon.

Trigonometry of the eclipse

Let us recall the famous sine relation - as it may be found in any high school textbook on trigonometry:

Figure 7

The next drawing shows the mathematical triangle associated with the parallax angle alfa:

Figure 8

If we apply the sine relation formula to our astronomical triangle, we find:

Figure 9

Within a very few percent, the distance from Europe (Observer) to the Moon is equal to the geocentric distance.

The formula above may thus be reduced to :

Figure 10

Math Exercise: Repeat the method above, and do all the needed reductions.

The distance to the Moon

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

The geographical latitude of Ystad (Sweden) is +55.43 deg. The Lunar declination was -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 apply our distance formula.

At the beginning of these notes, we found that the Sun-Moon distance, as seen in the sky, `ought to be' far too high. According to above, alpha `should 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 similar to this one (Tycho Brahe Planetarium - nearly same latitude):

Figure 11. Click on image to see larger version (JPG, 18k).

Thus, the observed distance in the sky between the geometrical centres of the Moon and the Sun is significantly reduced.

The Ystad group observed - similar to many other Astronomy On-Line groups - the fraction between h and W:

Figure 12. Click on image to see larger version (GIF, 20k).

This fraction depends on when you observe it - as demonstrated most beautifully by the Kuffner Group:

Figure 13

However, during the maximum itself the h/W fraction becomes - in a first and reasonable approximation - a useful tool when estimating the angle alfa above.

Figure 14

As we show here - there is a simple formula - which converts the observed h/W value into a center to center distance. Applying this formula, it is possible to show that the observed center-center distance of our two objects is 0.16 Deg.

Figure 15

Put in another way, when seen from Ystad in Sweden, the Moon is shifted with a parallax equal to alpha = 0.95 Deg - 0.16 Deg = 0.79 Deg.

In addition - show that the angle "Geographical Latitude minus Declination" gets equal to 62.1 Deg.

Now try this exercise: Use your pocket calculator. Show that the sine relation 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 390406 km.

So this method is really quite promising!

The method has one advantage : All calculations above may be done with a pocket calculator. However - please observe a qbasic program has been published - performing all these calculations - interpolations - etc.

In addition - some people have asked : "Why does this method work at all ?". Usually eclipse-math require tough skills in advanced 3 D spherical trigonometry - or advanced vector math - as described in the recent Comet Hyakutake EAAE Trans-European parallax project.

However, because this eclipse is rather close to the meridian - and because maximum happens approximately within North-South direction, see the Ystad picture above (Figure 14) - the advanced 3D math may be approximated with this more simple 2 D method.

Author:Mogens Winther
Soenderborg Amtsgymnasium
Soenderborg Denmark

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