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# Celestial navigation - Part 1 Latitudes.
Let us pose the question, how do we find our position on Earth? The ancients of the past and the trailblazers used stars to help guild them, long before GPS were sent into orbit. Let us see how this works!
To start, one would place some kind of imaginary coordinate system on Earth, and indeed we do, they are the latitudes and longitudes (which is essentially **spherical coordinates** with a fixed radius, that one encounters in multivariate calculus!). We shall define them and begin with **latitudes**. We will assume that the Earth is a **perfect sphere** (which it is not), but it will be a good start.
## The Equator.
Yes, the Earth is **round** and roughly spherical in shape, and it spins on its **rotational axis**. If we draw out this imaginary axis, it meets the Earth at two points, the **(true) north pole** and **(true) south pole**. The plane that is perpendicular to the rotational axis and also halfway between the two poles defines the **equatorial plane** as well as the **equator** of Earth.
![[1 teaching/summer program 2023/week 2/---files/Finding_distances_using_sunrises 2023-05-12 13.17.39.excalidraw.svg]]
%%[[1 teaching/summer program 2023/week 2/---files/Finding_distances_using_sunrises 2023-05-12 13.17.39.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Finding_distances_using_sunrises 2023-05-12 13.17.39.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] In above diagram, the Earth rotates in a **right handed** direction (the direction of your fingers as you curl them on your right hand, with your thumb pointing up). Is this correct ? How can you convince yourself that this is the case ?
## The Latitudes.
The latitude of a location $L$ on Earth is the measure of the angle $\theta$ formed by a ray drawn from the center of Earth to the equator and a ray from the center of Earth to $L$. We often report the latitude as a positive value, and add the suffix $N$ for northern hemisphere or $S$ for south hemisphere relative to the equator. So Vancouver, Canada has a latitude of about $49.3^\circ N$, while Buenos Aires, Argentina has a latitude of about $36.6^\circ S$. However, we also use the convention that latitudes in the southern hemisphere to take on a negative value, in this case we do not need the north or south suffixes. A zero degree latitude would be the equator. **Locations in the same plane parallel to the Earth's equatorial plane would have the same latitude.**
![[1 teaching/summer program 2023/week 2/---files/Finding_distances_using_sunrises 2023-05-12 10.48.53.excalidraw.svg]]
%%[[1 teaching/summer program 2023/week 2/---files/Finding_distances_using_sunrises 2023-05-12 10.48.53.excalidraw|🖋 Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Finding_distances_using_sunrises 2023-05-12 10.48.53.excalidraw.dark.svg|dark exported image]]%%
![[---images/---assets/---icons/question-icon.svg]] Using the north and south suffix, we only need from zero to what degree to be able to describe all locations on Earth latitude wise? If we have a latitude of $146^\circ N$, what latitude is it equivalent to ?
![[---images/---assets/---icons/question-icon.svg]] Suppose location $A$ is **directly north** of $B$ on Earth, where their latitudes differ by $1^\circ$ . **How far apart are these two locations**? That is, if you were drive from $A$ to $B$, how far would you have driven? Your answer should depend on the radius of Earth $R_\text{Earth}$, which you can take it as about 3960 miles or approximately 4000 miles.
For historical reasons, angles were often reported in **Sexagesimals**, or **base 60**. So $1$ degree is divided into $60$ arcminutes, and $1$ arcminutes is divided into $60$ arcseconds. The short hand for $1$ arcminute is $1'$, and $1$ arcsecond is $1''$.
![[---images/---assets/---icons/question-icon.svg]] Des Moines, Iowa ($41^\circ 39' 23'' N$) is approximately due north of Springfield, Missouri ($37^\circ 6' 58'' N$). How far apart are these two cities?
![[---images/---assets/---icons/question-icon.svg]] Suppose we have two locations on Earth, **one directly north of another**, with latitudes $L_{1}$ and $L_{2}$ in degrees (using the negative latitude convention if it is in the Southern hemisphere). Express the distance between the two locations $d$ as a function of $L_{1}$, $L_{2}$ and $R_\text{Earth}$, the radius of Earth.
![[---images/---assets/---icons/question-icon.svg]] Suppose you stand on the equator on Earth, which has zero latitude. As the Earth rotates on its rotational axis, you would spin around on circle of radius $R_\text{Earth}$, the radius of Earth. Now suppose you stand at latitude of $L$ degrees (using negative if Southern Hemisphere), as the Earth rotates on its rotational axis you now spin around on a circle of a smaller circle of some radius $R$. Write $R$ as a function of $L$. (Draw a diagram to help you!)