Date modified: Mon 2023-09-18 . 09:01 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Cosmic distance ladder : The size of the moon. Ok let us keep going, how about the size of the moon? To do this we will use the **curvature of Earth's shadow** on the moon ! Imagine the situation when the moon enters and leaves Earth's shadow during an eclipse. Let use try to find how big that shadow is in comparison to the moon. When viewed from the Earth to the moon, the moon moves into a circular shadow casted by the Earth: ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.38.24.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.38.24.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.38.24.excalidraw.dark.svg|dark exported image]]%% Let us assume that the size of Earth's shadow is roughly the same as Earth's size. Now we just need to compare the size of the moon with Earth's shadow. Observe that the circular shadow of Earth on the moon has a curvature (it is not a completely straight line). Let us consider the situation when the edge of Earth's shadow just touches the center of the moon, like so: ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 15.15.09.excalidraw.svg]] ![[---images/---assets/---icons/question-icon.svg]] Let us re-orient this, and connect the two points where the circles meet and draw a straight line. Call the distance of this line to the moon's center $f$. Let us also denote $R_\text{moon}$ and $R_\text{Earth}$ as the radii of the moon and Earth respectively, **write a relation** between $f$, $R_\text{moon}$, and $R_\text{Earth}$. ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 15.23.43.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 15.23.43.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 15.23.43.excalidraw.dark.svg|dark exported image]]%% If you do it correctly, the relation should look relatively uncomplicated. Hint: You can do this geometrically using Pythagorean theorem, or you can try to **coordinatize it**. That is, imagine the moon circle is centered at the origin, and the Earth's shadow's circle is centered somewhere else, write out their **equations** in the $xy$-plane, and solve. === ## A real image of Earth's shadow's curvature on the moon. Here is an image taken from Wikipedia, showing the curvature of Earth's shadow on the moon, where the shadow touches about the center of the moon: ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 15.03.05.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 15.03.05.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 15.03.05.excalidraw.dark.svg|dark exported image]]%% Abstracting it allows us to measure it : ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.24.33.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.24.33.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.24.33.excalidraw.dark.svg|dark exported image]]%% ![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.28.16.excalidraw.svg]] %%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.28.16.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder_4 2023-05-10 14.28.16.excalidraw.dark.svg|dark exported image]]%% ![[---images/---assets/---icons/question-icon.svg]] Now, use all the information we have so far, **estimate the size of the moon !** #summer-program-2023