On the Sizes and Distances of the Sun and Moon -
The Ancient Astronomers Aristarchus and Eratosthenes used geometry
to find the distances to the Sun and Moon. Aristarchus noted that the moon was just barely able to cover the sun during a total solar eclipse, meaning that the two have the same angular size, or apparent size, even though they are at different distances from the Earth. This means that the actual sizes of the moon and the sun could be used to determine their distances by the use of proportions.
distance to moon / Distance to sun = Actual size of moon /Actual size of sun
Aristarchus (310 - 230 BC) determined the Size of the Moon
a. During a lunar eclipse, the Moon travels through the Earth's shadow.
b. We can see that the Moon is smaller than the Earth's shadow so it must be smaller than the Earth.
c. Aristarchus judged that the Moon is about 1/3 the size of the shadow, so the Moon must be 1/3 the size of the Earth.
d. (Actual ratio is REarth = 3.7 RMoon.)
Aristarchus determined the distance to the Moon
a. The Moon subtends an angle of 0.5o (Half of the width of your little finger held out a arm's length.)
b. D = Diameter of Moon (Aristarchus knows this!)
c. Theta = angle subtended by the Moon
To find the distance to the Moon:
1) Angle in radians = 2 pi x (angle in degrees)/(360)= 2 pi x 0.5/360 = 8.7 x 10-3 radians
2) Distance = d = D/(angle in radians) = D/(8.7 x 10-3) = 3.5 x 103 km / (8.7 x 10-3) = 4.0 x 105 km
(Actual distance is 3.8 x 105 km) Another way the Greeks were able to determine the distance from the earth to the moon comes from a concept called parallax, which is a sort of triangulation. This means that a stationary object appears to move if an
observer changes his position. You can test this idea by placing a pencil at arms length in front of your eyes and directly in front of your nose. Close your left eye and note the position of the pencil. Now, at the same time open your left eye and close your right eye. The pencil appears to have shifted position. We can pretend that the moon is the pencil, and that each eye represents an observer. If we can find the angle that the moon makes with one observer (one eye), we can use that angle to find the distance of the moon using the following equation: Distance to the moon = (1/2 distance between observers) * (tangent of the angle between the observer and the moon)
Modern method for finding distance to Moon:
1) Astronauts left a mirror on the Moon.
2) We aim laser at the mirror and time how long it takes for the laser to bounce back to the Earth.
3) Laser light, travels at the speed of light.
4) 2 x distance = c x time
Aristarchus' Method for finding Distance to the Sun
1) Once the distance to the Moon is known, use trigonometry to find the distance to the Sun.
2) When Moon is in the first quarter phase, the angle between Sun and Moon (measured from Earth) is not exactly 90 degrees.
3) Measure angle, and then
§ dsun = distance to the Sun
§ dmoon = distance to the Moon
§ dmoon = dsun cos(angle)
§ Aristarchus measured the angle to be only 87o and found that the Sun was only 20 x further away from the Moon, but he had the right idea.
§ In 1630 Vendelinus used a telescope and measured an angle of 89.75o which would make the Sun 230 x further away than the Moon, which is a bit better than Aristarchus' measurement.
§ The actual angle is 89.853o Therefore the Sun is 390 x further away from us than the Moon.