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For a landing via suicide burn, I have identified three phases:

  1. Starting at $t_0$, the craft approaches the ground with $v_x$ constant and $v_y$ gradually increasing (due to gravity)
  2. At $t_b$, the craft accelerates in retrograde direction until the speed vector $(v_x,v_y)$ is below a certain safety margin.
  3. The craft maintains the speed inside the safety margin until it touches down at $t_1$

I am trying to optimize $t_b$ for fuel consumption.

Obviously, there are three cases to consider:

  1. $t_b$ might be too early, causing the fuel to run out and the craft to crash
  2. $t_b$ might be too late, causing the craft to crash due to a high residual velocity
  3. $t_b$ might be inside a safe landing corridor

In order to optimize this, I need a goal function, i.e. some kind of score that I assign to each outcome. The case 3 is trivial: Here we can attempt to maximize the residual fuel when landed. In case 2, I can probably minimize the residual velocity.

But what about case 1? Intuitively, if the lander has enough power to brake to a standstill (and something), residual velocity should be smaller for larger $t_b$, but is that true?

Dan Pichelman
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choeger
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  • Isn't the whole point of a suicide burn that you hit $v_x=0, v_y=0$ at exactly the moment of touchdown? At that point, you don't need any additional fuel. So for a suicide burn, you'd optimize for minimum residual fuel when your velocity and altitude are both zero. – user May 02 '17 at 20:11
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    @MichaelKjörling That's fine for idealized mathematical solutions, but in practice you want to end the maximum-thrust suicide burn phase with some downward velocity at some positive altitude with enough remaining fuel to manage the rest of the descent incrementally. – Russell Borogove May 02 '17 at 20:20
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    The optimum case is trivial--burn at full throttle until the engines shut down due to starvation at the instant you touch the target. In practice you will back off from this a bit as none of the factors involved can be measured with extreme precision. – Loren Pechtel May 02 '17 at 20:32
  • What you are describing is not a suicide burn it's a landing. The most fuel-efficient landing is the suicide burn. – Roman Reiner May 02 '17 at 20:53
  • @LorenPechtel I wouldn't call starving the engines optimal, since it could be preferred to land with as much remaining fuel (for a return trip perhaps?) as possible. Nor would I call finding time of ignition for this kind of burn a trivial task, as it pretty much boils down to the problem of optimizing a powered trajectory under inverse-square gravity field. – Przemek D May 10 '17 at 13:27
  • @PrzemekD I was talking about the case where the engines won't be used again--I'm not aware of anything landing on anything bigger than an asteroid for which that was not true. The Falcons land and then are hauled off for inspection, any fuel remaining when the Apollo landers touched down was abandoned--the ascent stage had it's own engine and fuel and could not use the descent fuel. – Loren Pechtel May 10 '17 at 14:01
  • @LorenPechtel The landings you mentioned were not optimized for fuel consumption but for safety - and for neither objective is starving the engines on touchdown optimal. I'm only arguing the valid thing to say would be "optimum case is to burn at full throttle until the engines shut down at the instant you touch the target". – Przemek D May 11 '17 at 05:42

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