Bonus challenge: Celestial Navigation

CHALLENGE for designing a working method for Celestial Navigation

Difficulty: Difficult 

Requires entry-level college maths. Programming knowledge will help (but is not necessary). 
Cost: $$$

This challenge will require dedicated work for weeks/months. 

Celestial navigation has been a tool for navigators for a very long time. Stars, the Sun, Planets, and the Moon have been reliable beacons in the sky and helped many sailors find their way over the oceans and to port. 

There is however a rather strong emphasis on a spherical model of the Earth for most of the celestial navigation practiced today. The method used since about 1830 (Multiple star-fixes and the intercept method. Longhand haversine reduction formulas etc.) is based on Eratosthenes' formula which predicts a 1-degree altitude shift for any celestial object for every 111.1 km traveled towards its Geographical Position (GP).
The GP is the unique point on Earth where the celestial object is straight overhead (in zenith). 

Now it is time to start working.

1. Study the subject of celestial navigation.
   This can be done in several ways. There are tons of books available, but a good start is to study the Wikipedia article which will give you a historical background. The next step could be to study material online, such as this Youtube course. The course will teach you all of the "old-school" tools and techniques used. 

2. Buy the appropriate gear.
   This includes a sextant, an accurate watch, a nautical almanac, sight reduction tables, and proper writing equipment. 

3. Start making observations (sights) and practice the method of sight reduction (finding your location). 

From this point, you can choose to go the "old-school" way (using paper, dividers and plotting equipment). Another option is to simplify the work of sight reduction and use a computer instead. For your help, I have made a simple script written in the programming language Python. You can find the script here, including some basic documentation. The script will rapidly make a sight reduction for you given two or more sights taken (with your sextant). You can even run the script on your (Android) phone if you install this app. This allows for a very portable setup, not depending on any internet connection. If you go through the documentation and the code you can see it is all based on spherical geometry. 

CHALLENGE: Your job is now to construct a re-make of the model used and convert it to a Flat Earth model (or any other non-spheric model). You can do it the "old-school" way or build software. It is your choice. 

• You have to decide on the used geometry. An Azimuthal Equidistant model, a.k.a "Gleason model" might be your choice. 

• You also need to decide the elevation of the Sun, Planets, and Stars.  For the conventional spherical model this is not needed (the celestial objects are almost infinitely far away) but in a model with local celestial objects (e.g a Flat Earth model), the elevation(s) must be known for the geometry to work out.

• As you can see the used Nautical Almanac (for 2024) is bundled in the project too. It contains the locations on Earth where every celestial object is straight overhead for any given time (the so-called Geographical Positions or GP:s). You need also to consider the validity of the Nautical Almanac. If you choose to dismiss the almanac you need to replace the values with some other kind of dataset.  

• You need to decide how to handle horizon drop. Maybe you will decide to skip this altogether for a flat earth. Your choice must be part of the method you devise. 

• You may also need to check how to handle atmospheric refraction. I have used Bennett's formula in my code but you may need to change this. 

• You need to decide on the structure of space and the laws of optics. Standard (non-relativistic) science assumes space can be described as a 3D Euclidian space with light moving in straight lines. Any deviation from this axiomatic structure must be carefully described by you. 

• Finally consider computational complexity and usability. A solution that requires fancy and advanced computer algorithms may be tempting, but remember that the classical spheric algorithm solves the sight reduction problem analytically, i.e. using a set of simple mathematical formulas. A standard solution should produce a sight reduction in an instant on a computer, and in a couple of minutes manually in the hands of a skilled navigator. Also, remember the practical complications of requiring sailors and travelers to bring computers. An algorithm that can be implemented by printed tables (sight reduction tables) without computers will be vastly superior. 

• Another route is to find someone else having done this work before you. Take a look in search engines for scientific publications.  Here is a good start. This can save you a lot of time.

If you succeed in producing a reliable and usable method (analog or digital) for celestial navigation using your alternative model and techniques then you have made a breakthrough in the research about the shape of the Earth. It would be best if you also tried it out in practical navigation at sea. The conventional (spheric) method of celestial navigation can produce sight reductions with an accuracy of about 1 nautical mile in the hands of a skilled navigator. Your method should be comparable or better regarding accuracy. Any breakthroughs of this kind should be reported to the naval/marine authorities of your country and/or the mathematical institution of your local university.

As a warm-up to this, you can also consider this easier challenge.

Finally: This is a difficult challenge. Maybe you need help. If so don't hesitate to consult others who may be able to assist in the more tricky mathematics. Also, consider how to fund your project. Crowdfunding may be the way to go. 

Good luck!