Is it equally fuel efficient to burn off horizontal dv before vertical dv?

[davidparks21]

Assuming you only burn your engines at full thrust (obviously it's wasteful to drift down slowly at half thrust, wasting fuel against gravity), and no atmosphere, would it be equally fuel efficient to burn off your horizontal DV first, let yourself drop straight down, then perform a vertical thrust to kill the vertical DV (as late as possible)? Or is there a cost savings to coming in at shallow angle and burning off DV directly against retrograde?

The former approach makes landings easier to target, but I ran out of fuel unexpectedly trying this on Tylo recently (burning off 2000 DV on a shallow approach against mountains is risky business). Perhaps I just underestimated my DV requirement (easy to do on Tylo), but I wasn't sure if this approach had cost me something in efficiency.

[Tsevion]

Owing to the Oberth Effect you want to burn as much as possible against (to slow down) or in (to speed up) the current direction of travel. The most efficient landing profile on a vacuum body is pretty much the exact opposite of the most efficient launch. As you probably are aware, the most efficient launch on a body in a vacuum is as much horizontal and as little vertical burn as possible. The most efficient landing is exactly that in reverse. You want to get your periapsis as close to the ground as possible. Then when nearing the ground, start burning horizontal. Burn just enough vertically to keep your vertical velocity at or near 0, as you slow down, you'll need to burn increasingly more vertically. When your horizontal velocity is gone, if you aimed close enough to the ground you should be not moving, and only slightly above the ground.

[numerobis]

It's much more efficient to burn off both simultaneously, for two reasons.

Reason the first: say you have x m/s of horizontal speed to burn off, and y m/s of vertical speed. If you burn them off one after the other, you need to burn x + y m/s. If you burn them simultaneously, you need to burn sqrt(x^2 + y^2) -- the hypotenuse of the triangle. The same argument goes for whenever you want to burn prograde *and* you have a normal and/or radial burn to do as well to change your orbit -- always do as many burns as possible in one.

Reason the second: it takes time to burn off that speed. During that time, gravity acts upon you and increases the amount of vertical speed you need to burn. Horizontal speed induces a centrifugal acceleration that reduces your vertical speed. So the sooner you burn off horizontal speed, the more vertical speed you have to also burn off.

To take off, the optimum is to burn mostly horizontal, but up just enough to fight gravity. Landing seems to be basically the same in reverse.

[tsotha]

I agree with Tsevion. Something like this:

[LethalDose]

The savings you'd see are likely more due to the Pythagoras than Oberth, but it depends on what altitude you'd be killing youre horizontal velocity.

Either way though, it's more efficient to make one burn at low altitude to land.

[cantab]

When you're thrusting horizontally, all your thrust is going into slowing you down. When you're thrusting upwards, some of it is going into fighting gravity. So the efficient way is to make a small deorbit burn, then a large braking burn that leaves you little distance to fall vertically.

Of course you do need to watch out for terrain. You can't have your approach passing through hills! A deorbit burn 1/4 of your orbit ahead of the landing site is usually a good compromise, depending on your orbital height.