(8d) How Distant is the Moon?--2
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: "How distant is the Moon?--2," section #8d: on disk Shipparc.htm, on the web |
Goals: The student will|
Terms: Solar eclipse (total and partial) (terms umbra and penumbra may be used by the teacher for total and partial shadow), baseline, parallax.
;Start by reviewing eclipses.
A lunar eclipse enabled the ancient Greeks to estimate how far the Moon was--over 2000 years ago. Such an eclipse tends to be long, and is often total, because the Earth is big and its shadow is much larger than the Moon, easily able to cover the Moon entirely, for hours.
A solar eclipse occurs when the Moon's shadow falls on the Earth, and is completely different. Because the Moon is small, its shadow will only cover part of the Earth.
You might think the shadow is as big as the Moon, but it is actually much smaller. If the Sun were a point-like object, the Moon's shadow would indeed have the size of the Moon. But actually, the Sun covers in the sky a disk of 0.5°, about as big as the Moon. As a result, only in a small region on Earth, maybe 100-150 km across, is it completely covered by the Moon. The Moon is close enough to Earth that if we move only a few hundred kilometers, our line of sight changes sufficiently to uncover some of the Sun, so that we only see a partial eclipse.
The Earth-Sun distance and Earth-Moon distance vary due to the elliptical orbits of the Earth and the Moon. Sometimes the Moon is too far to cover the Sun, and we do not get any totality. At such eclipses, even in the center of the shadow zone, a "ring of fire" is seen around the dark Moon, which is not quite big enough to cover the entire Sun.
This point is discussed in more detail in section 9c.]
Such a move essentially shifts the angle at which we observe the Moon. It only takes 100-150 km to move the Moon by a noticeable fraction of the width of the Sun's disk, so that part of that disk, covered during totality, is now exposed. Hipparchus, about 140 years after Aristarchus, realized that this too could tell us how far the Moon was, and would therefore give a way of checking what Aristarchus had claimed.
Let us try it here:
Go over section #8b of "Stargazers", without the mention of the 1999 eclipse, which should be discussed separately later, perhaps with the shortcut mentioned earlier. The questions below are for both lesson and review:
Guiding questions and additional tidbits
-- The Greek astronomer Hipparchus used an eclipse of the Sun to estimate the distance of the Moon. When did that eclipse happen?
-- What were the observations Hipparchus used?
-- Suppose for a moment that the eclipse happened when the Sun was right overhead. The event gives us a tall narrow triangle whose side is the approximate distance to the Moon. What would be the "baseline"--the base of that triangle?
-- If the Hellespont is exactly north of Alexandria, and its latitude is
9° more--in terms of the radius r of the Earth, what is the baseline?
So the distance covered by 9° is (9/360) 6.28 r = (6.28/40) r = 0.157r
-- What is then the angle at the top of the triangle?
From Alexandria, the same edge falls short of the edge of the Sun by 20% of the disk of the Sun. The disk covers 0.5°, so the edge of the Moon is shifted by 0.1°.
-- Suppose the Moon's distance is R. The baseline covers 0.1° in a circle of radius R, centered on the Moon. What is the length of the baseline, in terms of R?
So 0.1° equals (0.1/360) 6.2 R - (6.28/3600) R = 0.0017444 R
-- If the Sun is not straight overhead but (say) overhead at the equator, how does this change the calculation?
Present the 1999 eclipse--perhaps with other details, on the eclipse path. Can the calculation of Hipparchus be repeated here?
Shortcut to the solution: in 129 BC, Alexandria had 80% of the Sun covered while the Hellespont had totality, so the distance between the two corresponded to the angle subtended by 20% of the Sun.
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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Last updated: 12.17.2001