Would there be any advantages for a rocket with two "half stages"?

Question

A "half stage" is a rocket stage which separates with its engines, but not fuel tanks. Common examples include Atlas (1.5 stages), Atlas-Agena (2.5 stages), and Atlas-Centaur (2.5 stages).

On paper, one could create a "one-and-two-half-stage" rocket. It burns for a while, then drops some engines. Then it burns some more, and drops more engines. It then burns its remaining engines to orbit.

Alternately, a "two-and-two-half-stage" rocket is also theoretically possible. It burns for a while, drops some engines, burns some more, and drops its lower stage. The upper stage burns for a while, drops some engines, and then finished its burn to orbit.

I'm sure such a design would have numerous disadvantages (feel free to point them out in the comments). However, are there any advantages over other designs?

Related:

• Why not keep the engine for multi-stage rockets?
• Could the Atlas-B (SM-65B) have made it to orbit without jettisoning the first-stage engines?

Answer 1

A 1-and-2-half stage could have gentler acceleration curves than Atlas. That launcher, derived from an ICBM design, accelerated steeply to about 7g, dropped the booster section, then accelerated from 1.3g back to nearly 8g on the sustainer.

With two separate booster sections and a smaller sustainer, it could reduce its thrust in smaller steps, keeping the g-force peaks lower and the troughs higher, maintaining similar average acceleration while putting less stress on the payload.

The big disadvantage, of course, is that a booster section that can safely detach from a rocket in flight is heavier than the same engines would be if they were permanently attached. I note that Atlas’ booster masses about twice as much as a pair of H-1 engines while producing comparable total thrust.

Answer 2

One big advantage of the half-stage on first stage of the rocket - which is similar to the advantage of using short-burn SRBs as boosters, like with Arianne 6: high TWR early on.

High thrust-to-weight ratio early on is very desirable, significantly reducing gravitational losses, thanks to increasing the speed a lot early on: with higher initial vertical speed everything happens sooner: gravitational turn, orbital insertion - and that means way less gravitational losses. But that also means reaching MaxQ sooner - and in denser atmosphere, meaning it's "harder". In this case reduction of thrust through disposing of extra engines may be a viable option.

The problem with extra engines on 2nd stage is that it's usually light enough to be propelled by a single engine. In case it is not so, dropping the extra engines might be desirable to reduce TWR at the end of the burn (to protect the payload and rocket structure) - at no efficiency loss; throttled engine losses a significant fraction of specific impulse: less combustion products; lower chamber pressure; lower exhaust speed. Disposing of engines will not reduce efficiency, but will reduce thrust - and get rid of mass that is no longer needed at the reduced thrust. The practicality of that is questionable though: vacuum-optimized nozzles are big, and matter to engine efficiency - and often there's simply no room to squeeze more than one into the diameter of an upper stage. And you're still carrying the empty tank mass, no longer needed. Implementing these disposable engines as boosters with own (smaller) fuel tanks might be more practical.