None

[jeff-oh]

My apologies for wasting your time.

[raftingtigger]

Nice, but in my experience soil characteristic and depth (this often changes the soil characteristics drastically) has a much greater impact on holding power than the boom / stake ratio.

I can use a 24" boomstake in packed beach sand with the same boom as I use in medium soil with a 12" stake. The 12" stake will not even begin to hold in sand. Now if I dig the first 4" of sand away and get to tighter packed sand the 12" has a better chance of working. This is empirical data.

[Latherdome]

Longer booms exert more leverage at the juncture with the stake, requiring a much stronger/heavier boom to avoid deformation or fracture. With the boom length limited by strength/weight considerations to well below the ideal your analysis indicates, the stake itself must be long enough to penetrate to firm enough soil, or all is lost. These 2 considerations leave our booms much shorter than the stakes.

Orange Screws still hold better in most conditions, especially toward the soft end of soils. The special case where Boomstakes excel is in ground too hard to drive the Orange Screws into, including the super-compacted composite around many car-camping "tent pads" where you may be compelled to set up. In this case the booms may be superfluous, but they extend the useful holding capacity into a large overlapping range with Orange Screws in softer ground. Boomstakes pack smaller and are lighter than Orange Screws, too.

[Yarome]

It might be interesting to do an analysis of the boomstakes typically in use vs. standard stake styles. The "ideal" length ratios might give the highest holding power within the stake structure, but in general, not practical in implementation, packability, or versatile enough for the variety of soil conditions commonly encountered... IMHO.

Assuming the boom to be level with the ground, some additional, and fairly important factors would be the stake angle in the ground and the stake to boom angle.

[Yarome]

....the stake itself must be long enough to penetrate to firm enough soil, or all is lost.

...they extend the useful holding capacity into a large overlapping range with Orange Screws

...Boomstakes pack smaller and are lighter than Orange Screws, too.

Well said!

Under most common soil conditions, they are also easier to implement than OS's.

[jeff-oh]

All the above responses are outside of what the OP presented. Trying to go into why would greatly complicate matters. Instead of going down a rabbit hole. I'll let this thread die a natural death. Cheers.

[Latherdome]

Do have a look at these similar commercial anchors: https://deltagroundanchors.co.uk/collections/tent-pegs . Longer booms relative to stakes, yes, but necessarily in heavier materials, packing larger, and still with relatively shallow ground penetration.