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there's a meaningful payload to orbit advantage with hot staging that is conservatively about a 10% increase

(Elon Musk during Twitter Spaces interview on June 24th, 2023, timestamp 37:13)

In a line of work dominated by tiny incremental sub-cent percentage improvements, this ten percent figure is quite a lot. I can understand why there would be an improvement, I just don't understand how it's so much.

My understanding is that after separation, the second stage is in free-fall and is not experiencing the acceleration necessary to pull the liquid fuel towards the bottom of the spacecraft. In order to push the liquid to the bottom, either an ullage motor is used to provide some small acceleration (with the propellant being some kind of pressurised gas I assume) or a rubberised bag pressurised with helium is used in the fuel tanks themselves to push the fuel aftwards. Hot staging is another way of solving this problem without using the above methods, since the upper stage is still experiencing acceleration.

With hot staging, you get the mass saving of not having any ullage motors or such, but you need extra mass on the first stage to resist the blast from the second stage engines. You get to fire the second stage immediately rather than wait a few seconds for the first stage to drift clear. Again, I see how this could be an improvement, but I don't see how it could add up to so much.

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    $\begingroup$ the explanation could be that Musk's offhand remarks are frequently hyperbolic when referring to rockets, and outright lies when referring to other things $\endgroup$
    – Erin Anne
    Commented Nov 10, 2023 at 2:19
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    $\begingroup$ I think what may have happened is that they had planned to do some kind of tricky flip maneuver and use centrifugal force to spin off the second stage, similar to how they deploy Starlink satellites. This was to avoid the complexity of separation mechanisms or motors. Apparently they ran into problems with the flip maneuver idea (too much gravity losses?), but still wanting to avoid separation mechanisms they decided that hot staging was a quick solution. Which it seems to have been as they cranked it out amazingly fast, assuming that it works which we should find out sometime this month. $\endgroup$ Commented Nov 10, 2023 at 3:47
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    $\begingroup$ “Hot staging could increase payload by 10%.” Or it could increase it by 1%. Or decrease it by 3%. $\endgroup$ Commented Nov 10, 2023 at 3:54
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    $\begingroup$ In the interview Elon seemed to avoid stating what the 10% increase was being compared to, "there's a meaningful payload to orbit advantage with hot staging that is conservatively about a 10% increase in, well it depends on what it's being compared to but let's say in this case roughly 10% improvement in payload". Maybe the 10% increase is the improvement compared to a previous inefficient plan to flip the stack. $\endgroup$ Commented Nov 10, 2023 at 13:42

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The main benefit is that the second stage never stops accelerating, retaining close to maximum acceleration at all times.

One of most significant losses in rocket launches are gravitational losses; the fuel spent on keeping the rocket from falling down, as opposed to what's needed to accelerate it up and sideways. On top of whatever the rocket uses up to fly up, it constantly outputs extra 9.81m/s^2 equal to Earth's gravitational acceleration, to combat gravity. And until quite close to orbital speed, it needs to provide a large part of that, for all the time it flies. Thus the faster it reaches orbit, the less it loses to fight gravity.

Coasting during separation, the token acceleration of ullage motors, reduced thrust to allow the booster to clear the area undamaged, that all takes time, where Earth's gravity is stealing the speed from the rocket, costing extra.

There are a couple more much smaller factors: the booster acting as a thrust plate, giving starship a little "oomph" during separation, and thanks to Newton's 3rd law the Starship pushing the booster back down, saving fuel needed for boost-back - which can be used to accelerate Starship a bit more, the savings you've already mentioned, maybe a little more.

This all doesn't nearly add up to 10% of energy expended on launch though - and yet, the 10% extra payload mass is quite likely, due to the exponential nature of the Rocket Equation. TL;DR you need fuel to carry the ship, and its fuel. And you need more fuel to carry that fuel. And so on - meaning even small savings early on result in huge gains late in the flight. 10% of payload is nowhere near to 10% of the spacecraft mass, especially loaded with all the fuel. I don't know Starship's dry:wet mass ratio, but in typical cases it was of order of 90-95%, so 90 tons fuel for 10 tons spacecraft+cargo, say, 5+5 tons. If you manage to save 1 ton of fuel out of the 90, you get an extra ton of cargo, +10% total, +20% cargo.

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