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We all know that the Falcon 9 is designed for reusability, but there still have been some hull losses. Though this hasn't happened yet, say a Falcon 9 experiences an in-flight failure where four core engines shut down prematurely. Would it still be feasible to ignite the upper stage and attempt to reach orbit, while trying to land the booster?

My main concerns with this scenario are that the F9 isn't programmed to activate the center engine and two other random engines across from each other instead of preset engines, and that the booster will have more fuel that it normally would during a typical landing sequence.

NB: Cursory tests with Kerbal Space Program (RSS) at 10, 20, 30, 40, 50, 60 km altitudes (70 and 80 were ignored because those coincide with a normal MECO). Five attempts at a manual pad-return at each altitude resulted in:

Altitude Failures Successes Payload reached orbit?
10 5 0 No
20 4 1 No
30 2 3 No
40 1 4 No
50 1 4 Yes
60 0 5 Yes
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    $\begingroup$ Jettisoning the upper stage makes no sense. Delivering the payload is the primary mission. Landing the booster is only secondary. It's financially important, but not as important as delivering the payload. $\endgroup$ Sep 9 at 15:54
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    $\begingroup$ Many things are possible, so if you are asking if it's within the realm of possibility then yes. Realistically no. $\endgroup$
    – GdD
    Sep 9 at 18:14
  • $\begingroup$ @JörgWMittag Perhaps "ignite" would be a better word. $\endgroup$ Sep 10 at 0:21
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Say a Falcon 9 experiences an in-flight failure where four core engines shut down prematurely. Would it still be feasible to jettison the upper stage and attempt to reach orbit, while trying to land the booster?

It's unclear to me, given the phrase "attempt to reach orbit", if you mean discard the upper stage or separate it as usual. As Jörg W Mittag points out in comments, the upper stage and its payload are the important part; in the event of a multi-engine failure it would be preferable to burn all the first stage's fuel to get the second stage as high and fast as possible, and sacrifice the first stage.

My main concerns with this scenario are that the F9 isn't programmed to activate the center engine and two other random engines across from each other instead of preset engines

Programming isn't the issue, it's the TEA-TEB igniter fluid used by the engines. Each engine start requires a "dose" of that stuff to be carried, and I believe that only the designated boost-back and landing engines carry the fluid for restarts -- the TEA-TEB used for the initial liftoff comes from a ground equipment source. I don't know if it would be a significant weight/complexity hit to carry TEA-TEB for the other engines.

And that the booster will have more fuel that it normally would during a typical landing sequence.

This won't be a concern. The fuel tankage is shared among all the engines. The mitigation for engine failure on a multi-engine first stage is to keep whatever engines remain running at full throttle and use all the fuel in the hope of getting the second stage up. This maintains as much of the already-earned vertical speed as possible; even if the rocket is briefly under 1:1 thrust-to-weight ratio when engines fail, the weight is dropping quickly as fuel is consumed. Unless a lot of engines fail very early on, you'll still be climbing the whole way.

The recent Astra 0006 failure illustrated this dramatically; it lost one of five engines almost immediately on launch. It hovered just off the pad, and sideslipped for several seconds (away from the launch tower, fortunately), then finally started to climb as it lost fuel mass.

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