# Intro - The Earth is a Sphere? The question of Earth's geometry was always one that bugged our ancient ancestors. As a result, Erastothenes, an ancient Greek philosopher, devised an experiment to resolve the issue once and for all - he measured the length of the shadow of a stick # Latitudes and Longitudes Everyone lives on Earth. Well, almost everyone, and the people who aren't are still gravitationally bound to our little blue dot. Everyone on Earth occupies a certain point on our planet at every point in time - no person can ever fully "overlap". Feel free to crack this case, Dr. Frankenstein! While not so important for people, it's become important to mark important locations with a coordinate system. Today, we have something called the **geographic coordinate system** - or *GCS*. It's how GPS works - plus a few software corrections to prevent it from leading us into forests while on cars. ![[Pasted image 20240221215210.png]] *GPS - It's still pretty good though!* *Latitude* and *Longitude* represent the vertical and horizontal displacement from the **meridian** - the same one that passes through Greenwich. Going horizontally from the meridian gives us a *longitude*, which can go from a value of -180$^\circ$ (when facing east) and a value of 180$^\circ$ (when facing west). At the Greenwich Meridian, our longitude's 0. A vertical displacement gets us the *latitude*, which ranges from -90$^\circ$ (south) to 90$^\circ$ (north). The equator has a latitude of 0$^\circ$. You can imagine each one %%diagram! this is difficult to make.%% ## Colatitudes Further down this section, knowledge about something called the *colatitude* is going to help greatly. I won't spoil anything, but just note that it forms a part of an area on a sphere. Simply put, a colatitude is the difference in latitude of the North Pole (which has a latitude of 90$^\circ$) and the object. $colatitude = 90-\phi$ In our case, the latitude is written as $\phi$ - which'll stay constant until the end of time. ![[Pasted image 20240221222958.png]] *Bad colatitude graph - by me!* # The Azimuth If you've ever used a star tracking mount you've probably heard of words like "Equatorial" and "Alt-Az". Often times, most people call it a day here - alt az is objectively worse because it can't follow the Earth's rotation, yada yada ya. Another earth-based observation technique is the *horizontal* system - where instead of using global coordinates, a local system is set up around an 'observer' - usually a dot for simplicity. Not sure if that's a poignant metaphor for people these days! This system introduces the *altitude* ($h$), *zenith* ($Z$), *azimuth angle* ($Az$) and *zenith angle* (90$^\circ$ - $h$), which are used in tandem with the # Right Ascension and Declination Circles are cool. Being the 2D "no-edge" polygon, it's quite a challenge trying to figure out the area of one without $\pi$. Spheres aren't any different - after all, they're just circles which tend to bulge out. Even the formula for the surface area of a sphere's similar to that of a circle - just with an order of magnitude increase of 4. However, the bad part is that we *need* to know how to manipulate them - we need to know how to do maths along its surface, how to assign each infinite point on a sphere its own coordinate. Here's where *right ascension* and *declination* come in. Imagine latitude and longitude - just on a galaxy-wide basis, inclined with our planet's own orbital tilt. It's like a bigger sphere encompassing our little sphere! ![[Pasted image 20240406003819.png]] PERFECT. # Sidereal Time All of this, happening around us, and we still take the day for granted. What next, it'll abandon us and force us to use another scale? Uh oh. It seems that even time's grown fed up with humanity. There's something it's holding - what's it called? *Sidereal Time?* Sidereal time is ther # The Astronomical Triangle *Remember to refer to* [[Spherical Trig - Intro]]! It's time. The astronomical triangle's the same # Horizons For some reason, many people like to climb mountains. Despite the deaths on mountains, the thrill and the clout of climbing Mount Everest has only fuelled man's desire to chase danger and adventure. On top of Mount Everest, or on any high summit for the matter, the views are said to breathtaking. Fields upon fields, towns and cities, stretching as far as the eye can see. And hold on - the Earth's slowly bending... downwards? At every far corner, the Earth seems to buckle under its own weight, disappearing from view below the **horizon.** To some people, finding out where the *horizon* lies can be pretty useful - from the photogenic to the photographer, large fields of view can either make or break a photograph. Depth is pretty useful when you're up on mountains, and besides, the further you see, the more unique the experience is going to seem. That is, until you have to go back down. Mountain trash sure is a big problem! ![[HorizonDiagram.png]] *The Horizon - Visualised!* To us, the *horizon* is the limit to what we can see thanks to the curvature of the Earth. There are two types of horizon we need to deal with - the *astronomical* and *true* horizons. >[!Success]- Astronomical Horizon >Astronomical horizons refer to the horizontal line tangent to the observer! >[!Abstract]- True Horizon >True horizons, the one we see, are usually more complicated. With 'a bit' of geometry, we'll see that Note that $\theta$ is usually small. If we're an observer on the ground, we can use the cosine taylor series expansion to get an expression for the horizon distance; # Example Problems # What's Next?