If you are orbiting at a very high ‘circularized’ orbit, and have the ability to cope with high speed reentry, which is more fuel efficient (generally ignoring aerobraking):
Burn hard at apsis and let periapsis fall to the surface.
Or slowly lower the circularized orbit generally bringing lower the apsis and periapsis in many smaller burns?
When you say high speed reentry and ignoring aerobraking, do you mean that it’s fine to just hit the surface of the planet/moon at a super high speed? If so, it would be most efficient to burn hard at apoapsis, lowering the periapsis to below the surface. No point burning fuel to lower your apoapsis if you’re never going to get there again.
(Disclaimer: I learned just the basics of orbital mechanics in Honors Physics a few years ago. My more recent knowledge comes from Kerbal Space Program, and may be inaccurate.)
How long do you have? Most fuel efficient would be to burn hard at apsis, and let periapsis fall to the upper atmosphere, then use braking there over many orbits to reduce your eccentricity, mostly just lowering your apsis until you’ve lost enough energy to land.
(both periapsis and apoapsis ARE an apsis. )
When you burn hard at apoapsis , the burn down at the surface is just like a burn to change periapsis, so both apsis then shift to approx 0 altitude. (rather approx
)
So as long as the smaller burns are at periapsis or at apoapsis, thats the same energy cost as single burns. (apart from the cost to start and stop the thruster, and the loss of efficiency if you don’t run at the most efficient thrust.)
When you change the location (apart from the height) of the periapsis, or apoapsis you are burning fuel to “hover” or change orbit.
If you’re ignoring aerobraking, and landing on a smooth surface, then I suspect the ideal is to do a short burn to bring your periapsis to ground level, and then a hard burn at periapsis to bring your velocity to zero. This makes maximum use of the Oberth effect, where burns are most useful deep in a gravity well.
Not a maneuver for the weak of heart, though :).
If your starting orbit is sufficiently high–over 12 times the radius of the planet you’re landing on–then it will be more efficient to use a bi-elliptic transfer, where you first raise your apoapsis, then when at apoapsis lower your periapsis, and then finally circularize at periapsis.