Lots of civil engineers on the board, apparently. I'm civil/structural. Given enough bandwidth, I'm sure we could collectively come up with a design and calculations to rival the moon landing
But basically, this is a simple flag pole type foundation. As someone mentioned, they are much larger in diameter than you might first think. Your's are too small unless, possibly, they are set in stiff clay. Since we have no idea of the soil type here (sand, clay, loose gravel??), assume it only has a 'lateral' strength of about 30 PSF (pounds per square foot). That's not much. Think about it. If you pushed as hard as you could on the top of these tall poles/levers (they appear to be 8-10 feet tall), 30PSF resistance against the side of the foundation isn't much to overcome. So, as several have mentioned, you need larger diameter piers. I don't think depth is an issue. 4 feet is plenty - that's nearly half the pole height! But I would go with minimum 18-inch diameter hand-dug piers (and maybe 24 if you have loose soil). And yes, lose the Sonotubes.
If you really want to do it right, use a corrugated steel ground sleeve (with a plate welded to the bottom) in the middle of the hole before pouring the concrete. Pour the concrete around the sleeve, leaving the sleeve hollow/open. Then you set the pole inside the sleeve and tamp sand down around it. This makes it easy to center, plumb, and adjust the pole. At the top, you take a short piece of Sonotube and set it so it's a couple of inches above ground, with the top sloped for drainage. This gives a finished appearance and keeps water away from the pole. Nearly all flag poles are done this way.
