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# Cosmic distance ladder : To the Sun!
What about the sun ? Can we find out how far it is ? Aristarchus famously wrote about how one could find this distance, using a realization about **moon phases** !
## Half-moon phase.
![[---images/---assets/---icons/question-icon.svg]] During a half-moon, one half of the moon is illuminated by the Sun. What would be the configuration of the Sun, Earth, and the moon during a half-moon phase?
![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_3 2023-05-10 13.15.48.excalidraw.svg]]
%%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_3 2023-05-10 13.15.48.excalidraw|🖋 Edit in Excalidraw]], and the [[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_3 2023-05-10 13.15.48.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] In above diagram (not to scale), the Sun, Earth, and moon form a triangle. Convince yourself what kind of triangle this is. Label the distance from the Earth to the moon as $d_\text{moon}$ and the distance from the Earth to the sun as $d_\text{sun}$.
Now, denote angle $\alpha$ as the Sun-Earth-moon angle.
![[---images/---assets/---icons/question-icon.svg]] Write a relation among $d_\text{sun}$, $d_\text{moon}$, and $\alpha$, and solve for $d_\text{sun}$ in terms of the other variables. What piece of information do we need still ?
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## Lunar phases.
To figure out the Sun-Earth-moon angle $\alpha$, we will make the realization that there are in fact **two** half-moon phases, one the first-quarter moon phase, and the other the third-quarter moon phase. These two half-moon phases would have a configuaration of the following:
![[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_3 2023-05-10 13.44.29.excalidraw.svg]]
%%[[1 teaching/summer program 2023/week 1/---files/Cosmic_distance_ladder_3 2023-05-10 13.44.29.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Cosmic_distance_ladder_3 2023-05-10 13.44.29.excalidraw.dark.svg|dark exported image]]%%
So we just need the time between the third-quarter moon phase and the first quarter moon phase. You can look this time on the internet, or use the value roughly 13 days.
![[---images/---assets/---icons/question-icon.svg]] Knowing the time between the two half-moon phases, we can find $\alpha$. What is it?
![[---images/---assets/---icons/question-icon.svg]] Finally with everything found, estimate the distance to the Sun from Earth, $d_\text{sun}$.
Comment on your estimate with the known value from the internet. What are the absolute error and relative error (percent error) of your estimate? Does your estimate have the same order of magnitude as the known value?
What are the assumptions and estimations that could cause this difference in the values?
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