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Unformatted text preview: lated these using information from a lunar eclipse. Use Timothy Ferris’s composite photograph of a lunar eclipse (on the
web at http://web.mit.edu/8.901/images/moon.png) to make a similar calculation. The diagram below indicates the relevant geometry. You should make use of small angle approximations
where appropriate. Western Limb
1/4o Center of Solar Disk
Umbral Shadow 1/4o Earth Moon Center of Solar Disk
D i. Assume that only the darkest part of the Earth’s shadow (the umbra) corresponds to total
eclipse. Estimate the diameter of the circle roughly corresponding to the umbral shadow on
the composite image, and also the diameter of one of the lunar images. Note that the center
of the shadow does not lie on the line connecting the path of Moon’s center — why not?
ii. Compute the radius of the Moon compared to that of the Earth. Be sure to account for the
proper geometry of the umbral shadow at the distance of the Moon; for this purpose, you may
take the angular diameter of both the Sun and the Moon to be 0.5◦ . Estimate the uncertainty
in your answer, given the uncertainty in your estimate of the diameter of the umbral shadow.
Compare to the modern value of 6378/1738=3.67.
iii. Taking the angular diameter of the Moon to be 0.5◦ , calculate the Earth-Moon distance D
in terms of the Earth’s radius.
(c) Distance to the Sun. Aristarchus also estimated this. In the diagram below, the Moon at Q
is at ﬁrst quarter, so that the angle EQS is 90◦ . (Note that EQ is not perpendicular to ES .)
The interval from new Moon (at position N ) to ﬁrst quarter (at Q) is 35 min shorter than that
from ﬁrst quarter to full Moon (at F ). Given that the lunar synodic period (the interval between
two identical lunar phases) is 29.53 d, estimate the Earth-Sun distance (ES ) in terms of the
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