This got me interested to know if there is a gold standard for such calculations
For the orbital mechanics and geometrical aspects, the NASA JPL Development Ephemerides are commonly used. These have errors for the Sun and inner planets of tens of kilometers or less.
How to pronounce “Ephemerides”?
...and what is the range of error for it. For forecasts ...
This is not an answer.
Settings I used:
Results I got:
The kernel "pds/wgc/mk/latest_lsk_v0004.tm" was also used, but got clipped in the very last screenshot (it would be at the very bottom of the screenshot)
EDIT: As you note in the comments, "EARTH" means the Earth's geocenter. If you click on the help down arrow, it says
`SPICE name or numeric ID of ...
I'll use the atmosphere as my big natural lens. So I'll watch the upcoming mercury transit at the sunset time where the sun looks bigger than usual.
While the Sun and Moon might seem larger at the horizon, their angular size doesn't get larger. It's an optical illusion.
Will this make the black dot of mercury bigger or big enough to be obviously seen by ...
I did see the Venus transit before the Sun in 2004.
I used no telescope, but of course I used proper solar eclipse glasses to protect my eyes. The black circle was small but clearly visible.
But Mercury is much smaller, the transit is not visible without a telescope. See Wikipedia.
Here an image of a Venus transit:
Here the much smaller Mercury: