Can you show that timeanddate.com is based on a Round Earth model rather than any sort of pattern-based equation?
Yes! Very easily!
There is a lot of explanation on their page:
https://www.timeanddate.com/astronomy/about-sun-calculator.htmlOn this page:
https://www.timeanddate.com/services/api/astronomy-api.html you'll note that they offer software for professionals who need this kind of data.
Because I am one such person (I write flight simulator software for airlines and various military groups for a living) - I actually looked into using their software rather than writing my own. (We need to simulate sunrises and sunsets for our simulators - and to get things like the changes in these times due to your altitude to behave as they do in the real world).
When I looked into this, it seems that TimeAndDate.com use a software package called "SOFA" ("Standards of Fundamental Astronomy") that is provided by the International Astronomical Union (IAU). It's free to download and use, so lots of places that need sunrise/sunset (or Mars-rise/Mars-set or whatever) times often use it rather than writing their own programs.
So there was never a need for me to pay money for TimeAndDate.com when I could do what they do for almost no work.
Here is the documentation for SOFA:
http://www.iausofa.org/ http://www.iausofa.org/sofa_ast_c.pdfYou will see that this is a VERY serious piece of software - it takes into account a TON of complicated stuff like the effects of relativity and refraction of the air...it's exceedingly comprehensive.
If anyone here is a programmer - they can download SOFA for free from GIThub here:
https://github.com/abrudana/wwa/tree/master/WorldWideAstronomy/SOFAIf you're VERY smart - you could doubtless dig in and see the actual equations they use...but it's kinda daunting!
SOFA *clearly* uses RET equations (and the Heliocentric model) for the motion of all of the solar system bodies - from which they go on to derive things like sunrise and sunset times.
You can be 100% certain that they are using RET.
SOFA *works* - most astronomers use it for things like making sure that when they tell their telescope to take a 2 hour exposure photograph of some star or planet - that it'll move to the correct location in the sky and track that location in the sky to perfection.
If SOFA somehow didn't produce PRECISE positions for all of the objects in the sky - the world of international astronomy would be up in arms about it. Instead - there is a long list of presentations from the SOFA board and independent astronomers about how wonderful it is.
This cannot be wrong...it'll tell you when the center of the sun crosses the idealized horizon to within microseconds of precision...and if it was wrong to the slightest degree, we'd know about it.