The Magazine About Learning and Teaching Astronomy

Here are three articles with full details about Total Lunar Eclipses and how to measure the Moon's distance and size, from The Classroom Astronomer magazine, Issue 5, December 2010.   Shadow Cones.   In Earth's Shadow.  Caught! Shadow Reveals Moon's Size and Distance.

The Shadow technique can be done anyplace where the Moon can be watched through the beginning partial, total, and end partial phases of the eclipse.  It can be recorded by drawing or photography.

     The shadow of any spherical object in the solar system consists of a cone-shaped, dark central cone called the Umbra and an outer shadow zone surrounding the cone, called the Penumbra, where some sunlight shines directly into that space.  The umbral shadow edges may appear sharp to the unaided eye but it isn't as sharp as it seems.  This will cause some uncertainty in the measures, an important point all budding scientists should learn--no measurement is ever infinitely accurate.  We are attempting to measure the diameter of the umbral shadow at the distance of the Moon when it gets eclipsed this time.  The Moon's distance varies because of its elliptical orbit.  The Earth's shadow also varies; the cone can be longer or shorter depending on whether we are close to the Sun or farther.
  1.      The farther way the Moon is, the smaller the cross section of the umbra it cuts through.  So, measuring the ratio of the size of the moon in our sky to the size of the umbral shadow cross section gives us a way to find the distance to the Moon at that time.  We only need to measure the ratio (and size in degrees) of the shadow cross section that the Moon cuts through during totality.

Courtesy Wikipedia Commons

A summary of what to do to measure the Moon using the Shadow technique:

  1. Take drawings or photographs of the Moon as it moves through the Earth's umbra (central) shadow.  
  2. After the eclipse ends, put your photographs or drawings on a timeline along the edge of a piece of graph paper (preferably).  
  3. Here's the challenging part:  Using a compass and adjusting its size AND where the 'center point' should be, find the circle that BEST fits the circular edge of the shadow as the Moon moved through the umbra.
  4. Measure the diameter of your estimated shadow cross section in millimeters, and the diameter of a moon image, dividing the first by the second to get a ratio value somewhere between 2.5 and 3.0.  You will use this to find the Moon's distance.

Before The Eclipse

1. You MUST be within the zones where you can see all the total phases and at least some of the partial phases in order to do the Shadow Method.

2.  Download the Moon drawing sheet and copy it to paper. 

This is just an excerpt of the full sheet.  See the upside-down three-toed footprint on the right (west side)?  When you line up your drawings, that footprint should ALWAYS be to the right!

3.  Print also the timeline sheet attached to the drawing sheet and use or adapt this to another piece of paper.  The Moon moves its own diameter every hour so each hour mark MUST BE wide enough to just appear on the left and right edges of any Moon drawing image.

Extra Credit:

  1.      An interesting experiment to try near Sunset or Sunrise is to measure the ratio of the diameter of a spherical object (a tennis ball, a marble, a globe) and the length of its umbral shadow cone.  The umbra of the ball is easy to see on a screen when the screen is a meter or yard away from the ball but as the screen gets further away, the umbral fades into the growing penumbra (this is why we do this late or early in the day, we need the horizontal distance a shadow will go when the sun is low in the sky!)  One should find that, when you average several measures, you will find the ratio of shadow-cone-length to sphere-diameter is pretty much the same no matter what kind of sphere you use!  The ratio is just a bit over 100; a shadow cone is about 100+ times the size of the ball.  You can then scale this up to the size of Earth and calculate its shadow size, and even do so for the Moon, which should be just a little bit bigger in length than its average distance from the Earth.  This is why total solar eclipses are seen from such narrow strips. 

During The Eclipse

1.  You must begin drawing the edge and coverage of the shadow on the Moon circles just before the partial phases begins and end after the partial phases ends.  We suggest you make your observations every 30 minutes.  Your drawings should draw the shadow edge as it appears over the lunar seas that you see with the naked eye (or low-power binoculars) and with the shadowed part shaded in. It is very easy and tempting to draw the eclipse as it appears in the sky relative to the horizon but this would be wrong!!! Observers should rotate their sheet of paper with Moon images to match the Moon as it appears, particularly as to which Maria (seas) are "up".  Record the times of observations on a white part of the Moon circle.  We suggest you NOT cut out the circles until after the eclipse.

     (You can also take individual photos during the eclipse and print them and cut them out and put them on a timeline.  There are some photographers who can take a series of photographs on the same image (sheet of film or stored CCD image and show the circular outline of the umbral shadow.)

After The Eclipse

1. Cut out and put the Moon drawings on the paper with the time line. Mark your hours and half-hours on the timeline as appropriate for your time zone.  If you drew the Moon every thirty minutes you should have Moon images overlapping.  This is a good thing, you will be able to "average out" some of your drawing discrepancies.  

     Note that though the Moon's features may appear to rotate as the Moon moves across the sky, they actually do not and all your lunar seas should always line up exactly the same way on the timeline - the seas that seem to form a three-toed footprint should always be on the right side of each drawing.

2.  You should see that the edge of the Earth's shadow has a circular form, but you will only see part of the circle.  Your next job is to make the circle that best fits the visible shadow edge arcs.  This may require several attempts to get it done well;  you are doing a BEST FIT and it won't be exact or going through all parts of the arc perfectly.  We find that if you make tangents around the drawn large circular edge at several places, and then use a T-square to get perpendiculars to the tangent lines, you will find they roughly converse near a point.  Adjusting your drawing compass in size and center location place, you will eventually find your 'best fit' circle.

Here are the Moon circles placed on a timeline paper.  Notice that all of them have the "footprint" facing to the right (west).  Also, see the tangent lines along the edge of the shadow and the perpendiculars from them that helped locate the center of the umbral cone cross-section.

3.  Measure the diameter of the moon image with a millimeter edge, and measure the diameter of your best-fit shadow circle.  Divide the second value by the first; you should get a ratio between 2.5 and 3.0.  Calculate the diameter in degrees of the shadow circle by knowing the Moon is 0.5 degrees and multiplying that by the ratio you just determined (e.g. 0.5 x 2.7 = 1.35)

4.  Check columns 1 and 2 in Table 1S to find out how far the Moon is this using the size in degrees of the Shadow diameter.

5.  Find the Moon's diameter in miles using columns 1 and 3 of Table 1S.  

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