How do you figure out how far the sun is away from the earth, by compaing shadow lengths at 2 different points

Question

Early astronomers apparently used the method to calculate how far the sun was away from the earth with respectable accuracy. Basically I'm looking for the mathmatical formula, or laymans explanation.

Answer 1

During the summer solstice, no shadow is cast at the equator, but the shadow increases as you go north and south. The ancients knew this and by using triangulation and their knowledge of geometry, they were able to get the first estimate of the distance to the Sun.

Answer 2

Hi. The two sticks can be used to establish a good estimate of Earths diameter, as pointed out. The shadow of the Earth on the Moon's surface during a lunar eclipse can be used to establish a good estimate of the Moon's diameter. Once you know the Moon's diameter then you know how far away it is. If you notice that the Moon and Sun appear about the same size in the sky, you can anticipate the the Sun's diameter (about 400 lunar diameters) and the Sun's distance (about 400 times further away than the Moon) are related. Knowing this you can plug in different diameters for the Sun and compare with the known orbital time of the Earth to determine distance.

Answer 3

Check the link, which gives the derivation you're looking for. It involves a series of steps, resulting in deterimination of sizes of Earth, moon, sun, and distances between them. See "Size Scales of the Earth-Moon-Sun System"

Bugman is quite right, you cannot figure out how far away the sun is just by looking at a couple of shadow lengths on the ground. More steps are necessary. A critical measurement is the angle between the moon and the sun when the moon is EXACTLY at quarter phase (roughly 87 degrees), which was why an accurate determination of the distance to the sun was difficult.

Answer 4

I don't believe that's possible... I do know that Eratosthenes calculated the cirfumference of the Earth with respectable accuracy by comparing shadow lengths.

If you could measure the distance to the sun with shadows, then the nature of shadows cast by the full moon would be grossly different than those cast in the day.

Answer 5

Here is a great site to get you introduced to the concept of triangulation, also talks about parallax.
http://astro.gmu.edu/classes/a10594/notes/l05/l05.html

Discusses the difficulties in measuring large distances.

You need to use triangulation. You start by knowing the distance between your two measuring points on earth, as that gives you your base. You then need to know the angle to the sun from your measuring points. Then it is trigonometry to determine the distance.

The angle to the sun can be determined by the shadow cast from an known sized object. Assume because of the large distance between the sun and earth, the suns rays are parallel to each other at your two measuring points on the earth.

That should be enough to get you started.

Answer 6

The Earth, like numerous different planets, are being slowly pushed removed from the sunlight. 2 mechanisms clarify that: one is the rotation of the sunlight on its axis, which creates a coupling and momentum circulate between the sunlight (which slows down over the years) and the planets which %. up this lost rotational power as extra skill power making them pick the flow slowly (very slowly) out. For the checklist, the comparable coupling exists between the spinning Earth and the Moon, which additionally drifts away. the 2d mechanism is that the sunlight is dropping 4 million tonnes of mass (converted into power, as a consequence of fusing hydrogen into helium) each 2d. A gradually much less huge sunlight makes for a physique that doesn't carry its satellites in a relentless "grip".

Answer 7

Actually I agree with the 1st poster, I think they measured the circumfrence of the earth, not the distance to the sun.