Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Cosmic distance ladder : Size of the Earth Is it possible to measure large distances without actually measuring large distances? A famous story goes that the ancient Greek **Eratosthenes** was able to estimate the size of the Earth without, of course, actually measure out the entire Earth. Let's see how they did this! === ## The well in Syene. Eratosthenes heard of the story of a special **well** in Syene (in modern day Egypt), that on Summer Solstice at noon time, the well **casts no shadow**, and you can see the bottom of the well, illuminated brightly by the sun. ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 08.52.47.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 08.52.47.excalidraw|🖋 Edit in Excalidraw]], and the [[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 08.52.47.excalidraw.dark.svg|dark exported image]]%% ![[---images/---assets/---icons/question-icon.svg]] What can you conclude about the **position of the sun** in relation to the well on Summer Solstice at noon? Think about how light affects shadows. ![[---images/---assets/---icons/question-icon.svg]] If we draw out the Earth and indicated the well at Syene on its surface, **draw a beam of light ray from the sun (somewhere far away) to the well** that best describes Summer Solstice at noon hitting Earth. ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.10.39.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.10.39.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.10.39.excalidraw.dark.svg|dark exported image]]%% ![[---images/---assets/---icons/question-icon.svg]] Now, since the sun is so far away from Earth, the light rays from the sun to Earth will all be roughly **parallel** to each other. So on the diagram above, **draw a few more light rays from the sun** on Summer Solstice at noon. === ## Eratosthenes in Alexandria. Eratosthenes learned of this story of the well in Syene, and decided to investigate. Eratosthenes lived in **Alexandria** at the time (also modern day Egypt) which is about 5000 stadia away from Syene (**roughly 500 miles**). Eratosthenes learned of this from traders between the cities. ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 12.12.48.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 12.12.48.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 12.12.48.excalidraw.dark.svg|dark exported image]]%% Let us place Alexandria in our Earth diagram : ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.24.17.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.24.17.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.24.17.excalidraw.dark.svg|dark exported image]]%% ![[---images/---assets/---icons/question-icon.svg]] In this diagram, indicate the 500 miles between the two cities. Is this distance the distance of a straight line on the surface of the Earth? Using the **arclength of a sector**, write an expression relating 500 miles with the radius of the Earth and some angle. Draw them on the diagram. === ## Eratosthenes' shadow. So on Summer Solstice at noon, Eratosthenes in Alexandria set up the following experiment. He took a gnomon (a ruler) and stuck it in the ground vertically up, and saw that it casts a **shadow**. ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.34.28.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.34.28.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.34.28.excalidraw.dark.svg|dark exported image]]%% Since it is Summer Solstice, the shadow was not that long, but a shadow nonetheless. It was about $1/8$ the length of the part the gnomon sticking out of the ground. **From all this Eratosthenes was able to deduce the radius of the Earth !** ![[---images/---assets/---icons/exclaim-icon.svg]] **How did he do it?** ![[---images/---assets/---icons/question-icon.svg]] To help us analyze, let us draw (an exaggerated) gnomon at Alexandria sticking up on our Earth picture, and the shadow is also indicated. **Draw sun rays** to show that this gnomon is casting this shadow. ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.46.38.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.46.38.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder 2023-05-10 09.46.38.excalidraw.dark.svg|dark exported image]]%% Some hints. (1) On the diagram, indicate the center of the Earth, and connect them to the cities. Here there are two angles that will be the same, can you indicate where they are? (2) How would you compute this angle? [^1] (3) What is the equation relating arclength $s$ on a circle of radius $R$ spanning an angle $\theta$ ? Can you derive this ? And how is it helpful in our situation? ![[---images/---assets/---icons/question-icon.svg]] What is the radius of the Earth you estimated in miles? ![[---images/---assets/---icons/question-icon.svg]] Now look up the radius of the Earth on the internet and compare it with your estimation. === [^1]: Practically, one can just measure this angle directly by drawing a $1:8$ right triangle and use a protractor. #summer-program-2023