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This will happen once per day lol. For a geostationary orbit anyway, as the orbital period is 24 hours.
The point, as you mentioned before, is that the nighttime/eclipse part of this period will be very short and the day very long. Our night lasts hours, a geostationary satellite's night is minutes (maybe a little over 1 hour for the longest ones).
This website calculates eclipse periods for satellites: https://www.satellite-calculations.com/Satellite/satellite_eclipse.htm Apparently it's a seasonal thing, like 3 months you get daily eclipses, 3 months you get no eclipse, then another 3 months on and another 3 off. The 3 months with eclipses are the around the equinoxes, so Feb-Apr and Aug-Oct.
The Equator isn't in line with the angle the earth revolves around the sun.
Yup that's what I gathered from this. Literally had the "duh" realisation as I was writing that comment.
Eclipses still happen once per day, when they happen, it's just also seasonal with no eclipses in the summer and winter.