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were the engines "allowed" to shut down as a result of running out of propellant? Or were they issued a shut down command with some propellant left?

Shutdown was commanded when the appropriate trajectory was reached.

Even in multistage rockets, rocket engines are not run to complete depletion of propellant. The fuel/oxidizer mix ratio must be closely controlled to avoid damage to the engine; if the fuel ran out early the mixture ratio would go ox-rich, increasing combustion chamber temperature. The "trickling" flow of the last little bit of propellant would also produce rough combustion, which could be damaging. As Tristan notes, the turbopump would also be catastrophically damaged if propellant flow trickled.

were the engines "allowed" to shut down as a result of running out of propellant? Or were they issued a shut down command with some propellant left?

Shutdown was commanded when the appropriate trajectory was reached.

Even in multistage rockets, rocket engines are not run to complete depletion of propellant. The fuel/oxidizer mix ratio must be closely controlled to avoid damage to the engine; if the fuel ran out early the mixture ratio would go ox-rich, increasing combustion chamber temperature. The "trickling" flow of the last little bit of propellant would also produce rough combustion, which could be damaging.

were the engines "allowed" to shut down as a result of running out of propellant? Or were they issued a shut down command with some propellant left?

Shutdown was commanded when the appropriate trajectory was reached.

Even in multistage rockets, rocket engines are not run to complete depletion of propellant. The fuel/oxidizer mix ratio must be closely controlled to avoid damage to the engine; if the fuel ran out early the mixture ratio would go ox-rich, increasing combustion chamber temperature. The "trickling" flow of the last little bit of propellant would also produce rough combustion, which could be damaging. As Tristan notes, the turbopump would also be catastrophically damaged if propellant flow trickled.

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source | link

were the engines "allowed" to shut down as a result of running out of propellant? Or were they issued a shut down command with some propellant left?

Shutdown was commanded when the appropriate trajectory was reached.

Even in multistage rockets, rocket engines are not run to complete depletion of propellant. The fuel/oxidizer mix ratio must be closely controlled to avoid damage to the engine; if the fuel ran out early the mixture ratio would go ox-rich, increasing combustion chamber temperature. The "trickling" flow of the last little bit of propellant would also produce rough combustion, which could be damaging.