angle to the equator of the Sun on sky at a given time Is there an easy way to tell at what angle the solar equator is wrt the horizon at any given time of the day? I would think that it rotates 180 degrees from sunrise to sunset, but not in a linear fashion. But solar noon should always be 0 degrees wrt the horizon right?
Depends on the precision you need.. The sun's plane of rotation coincides with the plane of rotation of the solar system.
And for the latter, there is already a name in astronomy, the Ecliptic. Depending on your latitude on the planet $\delta$, the ecliptic plane cuts through your local horizon in east and west at an angle of $\alpha = 90°-\delta - 23.5°$.
This is the maximum height of the sun at midday and the sun's height varies thus between 0 and $\alpha$.

I think this would then be the angle you're looking for.



Another recent answer pointed out the angle between solar axis of rotation and the Ecliptic is $7.25°$. Translating that into the angle unter which one sees the solar equator, it is still $\alpha$, up to a small correction factor of $- \frac{R_{\odot}}{1 AU} \sin(\delta + 23.5°)$ which can be maximally $-0.05°$.