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As I pointed out in the comments, the difference appears to be due to the use of predicted values of $ \Delta T $ rather than a value updated with the most recent data from the International Earth Rot …

SPICE kernels are available for many NASA (and other missions). A large collection is available from NAIF. As well as some archived datasets. These can be processed using the SPICE toolkit, which i …

You can use the signs of the $ sin A $ and $ cos A $ to determine the proper quadrant: The equations in the Explanatory Supplement to the Astronomical Almanac are slightly different than the ones abo …

The equation of the equinoxes only applies to the origin, so it is assumed $ \delta = 0 $. So $\tan \delta = 0, \sin \delta = 0 $, which means your second equation simplifies to $\Delta\psi \cos \epsi …

Now, why would we choose a coordinate system that constantly changes? It's because it makes celestial navigation easier. …

Degrees are far easier for humans to work with. If you say something is $ 90^\circ $ from the horizon, people know what you mean instantly. In radians you'd have to say $ \pi/2 $ or 1.57rad, and tha …

The number of details involved depends on how accurate you want to be. There are more than just rotations involved though, you also need to account for parallax, aberration, gravitational deflection, …

Meeus provides a fairly simple algorithm in Astronomical Algorithms. An example implementation is available in Rise and Set Algorithm. The code is fairly simple and I have reproduced it below. A cou …

IAU SOFA and SPICE are overkill for what you need, and would require writing a C to C# wrapper around them. An implementation of VSOP87 in C# is all you really need. This is what was used to create …

I kinda found a solution, unfortunately it's a few hundredths of a degree off from those specified by Horizons. And it requires you to define a custom kernel for the observer's reference frame, which …