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A study of lunar and solar
eclipses
Rosa M. Ros
International Astronomical Union
Technical University of Catalonia, Spain
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Goals
◼ Understand why the Moon has phases
◼ Understand the cause of Lunar eclipses
◼ Understand why there are Solar eclipses
◼ Determine distances and diameters of the
Earth-Moon-Sun system
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Vision of lights and shadows
◼ The Earth-Moon-SunSystem:Phases and eclipses
◼ Relative positions and shadows
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Activity 1: Model of the far side of the Moon
◼ 2 volunteers: one in the centre (the Earth) and
the other revolving around it (the Moon)
◼ Place the Moon facing the Earth and have it
revolve around the Earth by 90° and rotate itself
also by 90º. Repeat the process until the
starting position is reached
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Activity 2: Model with flashlight (Sun)
to explain the phases of the Moon
◼ 5 volunteers: one in the centre (the Earth) and 4
others to simulate the 4 phases of the Moon
with masks (1 completely illuminated, 2
partially illuminated and 1 completely dark)
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Distances and diameters of the
Earth-Moon-Sun system
Earth Diameter 12 800 km 4 cm
Moon Diameter 3 500 km 1 cm
EM Distance 384 000 km 120 cm
Sun Diameter 1 400 000 km 440 cm = 4.4 m
ES Distance 150 000 000 km 47 000 cm = 0.47 km
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Activity 3: Simulation of Phases of the Moon
◼ Direct the
small moon
of the model
to the Moon
and we can
see both with
the same
phase
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Activity 4: Illustration Errors
◼ Phases of
the Moon
depend on
the position
of the Sun
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Moon Phases and Eclipses
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◼ Lunar eclipses
only occur when
the Moon is full
Activity 5: Lunar Eclipses
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Activity 5: Simulation of a Lunar Eclipse
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Activity 5: A Lunar Eclipse
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◼ Lunar eclipses can
be visible to half of
the Earth (night
side)
Activity 5: A Lunar Eclipse
Rosa M. Ros
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◼ Solar eclipses occur
only when there is
a New Moon
Activity 6: Solar Eclipses
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Activity 6: Simulation of a Solar Eclipse
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Detail of a Solar eclipse
Rosa M. Ros
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◼ Solar eclipses
are visible only
in a small region
of the Earth
Activity 6: Solar Eclipse
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... we are feeling emotion!
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Observations
•A lunar eclipse when there is Full Moon
and a solar eclipse when there is a New Moon
•A solar eclipse is seen only in a small area
of the Earth
•It is very difficult for the Earth and Moon to be
"well aligned”, thus an eclipse does not occur
every time that there is New or Full Moon
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Finally ... as an example ...
◼ Next total solar eclipse in Spain: August 12, 2026
(last one 2004 in a different area)
◼ Each year there are between 0 to 3 lunar eclipses
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Distances and diameters in order
to visualize and better understand the
distances to the Sun
Earth Diameter 12 800 km 2.1 cm
Moon Diameter 3 500 km 0.6 cm
E-M Distance 384 000 km 60 cm
Sun Diameter 1 400 000 km 220 cm
E-S Distance 150 000 000 km 235 m
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Painting the Sun
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Activity 7: Making the large “Sun”
look like the small “Moon”
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If every month there is
a New Moon and a Full Moon …
Why there is not
a Solar eclipse and a Lunar eclipse
every month?
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Because …
The plane of the Earth around the Sun and
the plane of the Moon around the Earth
are not in the same plane.
Both planes are inclined by 5º
and the angular diameter
of the Sun and the Moon is only 0.5º
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The eclipses only can take place if
the Sun and Moon are close to the
line of intersection of
the two planes.
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Activity 8: “Flip page” eclipse simulator
1. Trim and number the pictures in order
2. Paste each picture on a spiral notebook
3. Turn the pages quickly to see the demonstration.
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Activity 9: Determination of the Sun’s
diameter - observations and measurements
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Activity 9: Determination of the
Sun’s diameter
We can establish the proportion
and calculate the Sun’s diameter
L = 150 000 000 km Earth-Sun distance, l = tube length,
d = diameter of the Sun on semi-transparent paper
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Activity 10: Aristarchus’s Experiment
310 to 230 BC
◼ Established relationships between the Earth-
Moon-Sun distances and their diameters (but
could not determine any absolute value). This had
to wait until Eratosthenes.
◼ 1) Distance of the Earth to Moon and the Earth to Sun
◼ 2) Radius of the Moon and of the Sun
◼ 3) Earth to Moon distance and the Moon’s radius
◼ 4) The Cone of the Terrestrial Shadow
◼ 4) Relate them all
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1) Distance Earth-Moon and Earth-Sun
◼ cos a = EM / ES therefore ES = EM /cos a
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Rosa M. Ros
1) Earth-Moon and Earth-Sun
Distances
◼ Aristarchus a = 87º
then ES = 19 EM
◼ Now a = 89º 51’
therefore ES = 400 EM
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Rosa M. Ros
2) Radius of the Moon and of the Sun
◼ From the Earth, lunar and solar diameters
are observed to be equal to 0.5°
◼ Therefore, the
radius is
◼ Rs = 400 RM
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◼ Moon’s diameter from the Earth is 0.5 º
◼ With 720 times this diameter, we can calculate the
circular trajectory of the Moon
◼ 2 RM 720 = 2 p EM
◼ EM = 720 RM/ p
Rosa M. Ros
3) Earth-Moon Distance and Moon’s Radius
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◼ By analogy
◼ ES = 720 Rs/ p
3) Earth-Sun distance and Sun radius
Aristarchus’s
1st Heliocentric
model
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◼ In a Lunar eclipse,
Aristarchus observed that
the time required for the
Moon to cross the shadow
cone of the Earth was twice
the time necessary for the
surface of the Moon remain
covered (i.e. 2:1)
◼ It is actually 2.6:1
Rosa M. Ros
4) Cone of Terrestrial Shadow
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5) Relate them all
◼ (x+EM+ES)/Rs = (x+EM) / RE = x/(2.6 RM)
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Solving the system shows
(everything related to Earth’s radius):
◼ RM = (401 / 1440) RE
◼ EM = (401 / (2 p) ) RE
◼ Rs = (2005 / 18) RE
◼ ES = (80200 / p) RE
◼ If we assume RE= 6 378 km then
◼ RM = 1 776 km (actual 1 738 km)
◼ EM = 408 000 km (actual 384 000 km)
◼ Rs = 740 000 km (actual 696 000 km)
◼ ES = 162 800 000 km (actual 149 680 000 km)
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Rosa M. Ros
Activity 11: Eratosthenes’ Experiment
280 to 192 BC
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Activity 11:
Eratosthenes again
◼ Two cities on the
same meridian
◼ Simultaneous
observations
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Different shadows …
◼ Then the Earth is a sphere!
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Rosa M. Ros
Activity 11:
Eratosthenes again◼ p = p - a + b + g
◼ therefore g = a - b
where a and b are measured in radians
(180 degrees = p radians)
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Rosa M. Ros
Activity 11:
Eratosthenes again
◼ We measure the length
of the plumb line (or
stick) and its shadow
a = arctan (shadow)/(stick)
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Rosa M. Ros
Activity 11:
Eratosthenes again
◼ by proportionality
2p RE / 2p = d / g
◼ is deduced
RE = d/g
◼ g we know (in
radians)
g = a - b
◼ d is the distance
between cities -
using a map
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Rosa M. Ros
Our results with the
method of Eratosthenes
◼ Ripoll- Barcelona
◼ a = 0.5194 radians
◼ b = 0.5059 radians
◼ g = 0.0135 radians
◼ d = 89.4 km
◼ RE = 6 600 km (actual 6 378 km)
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Conclusions
◼ We now understand the eclipses
◼ Have established size relationships for the
Earth-Moon-Sun system
◼ It is verified that by observing and
analysing the data obtained, we can learn
much more about the universe
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Many Thanks
for your attention!