6683
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2011-02-01T00:00:01
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<mdoxml version="1.0"><ul id="stemLinks">
<li><a href="http://nrich.maths.org/2845">Warm-up</a></li>
<li><a href="http://nrich.maths.org/7268">Try this next</a></li>
<li><a href="http://nrich.maths.org/6480">Think higher</a></li>
<li><a href="http://motivate.maths.org/conferences/conference.php?conf_id=46">Read: mathematics</a></li>
<li><a href="http://plus.maths.org/content/dynamic-sun">Read: science</a></li>
<li><a href="http://eclipse.gsfc.nasa.gov/eclipse.html">Explore further</a></li>
</ul>
<div> </div>
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<p>A total solar eclipse is an amazing spectacle with the sun appearing to move exactly behind the moon for a few moments, with the view of the moon seeming to perfectly overlap the sun.<br></br>
<br></br>
But how perfect is this coverage? Is the view of the moon slightly larger than that of the sun or vice versa? In other words, exactly how perfect is a total solar eclipse as viewed from earth?<br></br>
<br></br>
To answer this question you will need to work out the apparent size or <a href="http://en.wikipedia.org/wiki/Angular_diameter">angular diameter</a> of the sun and the moon as viewed from Earth, using the following astronomical data: <br></br>
<br></br>
Radius of the sun: $695,500$ km<br></br>
<br></br>
Radius of the moon: $1,737$ km<br></br>
<br></br>
Radius of the earth: $6,371$ km<br></br>
<br></br>
Smallest distance between Earth and Sun $147,098,074$ km<br></br>
<br></br>
Largest distance between Earth and Sun $152,097,701$ km<br></br>
<br></br>
Smallest distance between Moon and Earth $356,375$ km<br></br>
<br></br>
Largest distance between Moon and Earth $406,720$ km<br></br>
<br></br>
<br></br>
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Here are some issues to consider:</p>
<ul>
<li>How much does the apparent size of the moon change, as its distance from the Earth changes?</li>
<li>How much does the apparent size of the sun change?</li>
<li>Can there ever be a perfect eclipse?</li>
</ul>
<br></br>
<p>Extension:<br></br>
<br></br>
Could other planets have perfect eclipses? To help you address this question you can consider the following <a href="/content/id/6683/MOONS%282%29.csv">data</a> for the moons of other planets in the solar system.<br></br>
<br></br>
</p>
<div class="framework"><br></br>
NOTES AND BACKGROUND<br></br>
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<br></br>
A solar eclipse seen from Earth is one of our solar system's most beautiful sights. From the moment the Moon begins to cross into the Sun's visible disc, the light covers the Moon in a special way, such that you have a feeling of the Moon's depth; the Moon as a near-sphere, rather than the disc we usually perceive. The process of the Moon slowly covering the Sun then takes a long time, as you
have a perception of this astronomical event. Then as the Moon just begins to cover the Sun there is an amazing "diamond ring" effect for a short time as the final rays of light are visible. Then for a few minutes the situation seems to remain still, and only at this time you can see a shimmering corona of particles that are around the sun, patterned by powerful forces, at millions of degrees
Kelvin. Then finally the diamond ring re-emerges, and the whole spectacle is seen in reverse. If you are lucky you might see one in your lifetime. Some people travel to wherever in the world one will be visible. Often people speak of the "eclipse coincidence", the fact that it is actually rather unlikely to have this spectacle on Earth, which we already believe to be quite a "lucky" planet,
having just the right raw materials and being just the right distance from a star of just the right age to support our life.</div>
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<mdoxml version="1.0"><span class="editorial">Well done to Dasom from Nanjing
International School for sending us</span> <a href="/content/id/6683/Dasom.pdf" class="editorial">this clearly
explained solution</a><span class="editorial">.</span><br></br>
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<h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/6683">This problem</a> gives a good opportunity to practise some trigonometry in context. Students have the opportunity to make decisions about how to model the situation and what diagrams to draw in order to address the questions raised. Students will need to take into account considerations of lower and upper bounds, and the effects of changing
measurements.</div>
<h3>Possible approach</h3>
<div>Perhaps start with some discussion about the apparent sizes of the sun and the moon, as viewed from Earth, and how the very similar apparent sizes makes solar eclipses possible.</div>
<br></br>
<div>Explain that the task is to work out the apparent sizes of the sun and the moon. In order to display this, students could cut out circles to represent the sun and the moon.</div>
<br></br>
<div>Ask students to consider in pairs what information they think they would need in order to work this out, and then provide any information they ask for from the data in the problem or using <a href="/content/id/6683/eclipse%20facts-2.doc">these information cards</a>.</div>
<br></br>
<div>It may not be obvious to students at first that they can work out angles to the edge of the sun or moon, or how to use this to compare the apparent sizes of the sun and moon. The diagram in the Hint may help.</div>
<br></br>
<h3>Key questions</h3>
<div>What assumptions do you need to make to model the situation?</div>
<div>Can you draw a diagram to represent the appropriate lengths and angles?</div>
<div>What difference does it make that the orbits involved are elliptical?</div>
<br></br>
<h3>Possible extension</h3>
Once students have had a chance to work out the necessary data for our moon, and drawn a diagram to show their findings, the problem provides <a href="/content/id/6683/MOONS%282%29.csv">data</a> for other moons so that students can investigate how rare a perfect eclipse is within our solar system. This task could be divided up between groups with each group looking at the moons of a different
planet, with everyone presenting their findings at the end of the lesson.<br></br>
<h3>Possible support</h3>
The problem can be scaffolded by discussing modelling assumptions together as a class and working out what sort of diagrams will be helpful - the image in the Hint could be used as a starting point for discussion.<br></br>
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This image (not to scale) represents the positions of the Earth,
moon and sun in a solar eclipse.<br></br>
<mdo:image height="287" width="648" alt="Earth, moon and sun" src="eclipse.png"></mdo:image><br></br>
Which lengths on the diagram do you know?<br></br>
How can you use this to work out the apparent size of the moon and
the sun?<br></br></mdoxml>
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<mdoxml version="1.0">Doug writes:<br></br>
Can we calculate a probability for that coincidence?<br></br>
<br></br>
Calculating the angle the Moon is seen to cover and the angle the Sun is seen to cover with simple trig, we find that they are very close, with a ratio of about 99% Moon:Sun.<br></br>
<br></br>
If we take the data for the other 22 moons in the solar system, as seen from their planet's surface, we find a spread of data from 0.1029 moon:Sun viewable angle for Mars' tiny moon Deimos to 26.53 for Neptune's large moon Triton.<br></br>
<br></br>
If we plot the 23 data points, there is no particular pattern that emerges, as we should expect. So we might reasonably assume a rectilinear distribution of $\frac{1}{26.53 - 0.1029} = 0.03784$ distribution height.<br></br>
<br></br>
So we have to ask what range of moon:Sun visible ratios would give this amazing effect?<br></br>
<br></br>
Our ratio is about 99%, so perhaps if we doubled the range, and allowed it to vary either side, we might assume this would be a reasonable range. This gives us bounds of 0.98 to 1.02, a range of 0.04. If we multiply this by our distribution height, we find that the probability of a moon:Sun ratio falling within this range is $0.03784 * 0.04 = 0.00151$.<br></br>
<br></br>
This of course treats all areas of the distribution as equal, and does not account for the "specialness" of the range we are looking at. So we should probably consider this probability an upper limit.<br></br></mdoxml>
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