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After writing the question Do Blue Origin's BE-3 engines need to run for 7 seconds to “warm up”?, I took a look at the YouTube video SSME ignition sequence (found in What are the small gas jets at the rims of the Space Shuttle Main Engines?) showing the Space Shuttle's main SSME engines, and they also take several seconds from the first appearance of burning exhaust to what looks like a stable, full thrust before the clamps are released and the shuttle begins to rise.

note: the connection is that both engines use LOX/LH2

What is happening here? How do the propellant flow rates and more importantly the resulting thrust vary versus time during this ignition sequence?

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There is a lot of good, detailed information about SSME startup and shutdown in "Space Shuttle Main Engine - the first 10 years Part 3".

The answers to this highly related question What are the phases of ignition (and flameout) of a liquid fuel engine? discuss many of the start parameters for the SSME but not the thrust.

This includes graphs of start parameters including chamber pressure (MCC PC).

enter image description here

Thrust is roughly proportional to PC. A thrust curve isn't shown in this paper, but here's a simplified one from my notes from working on the engine model in the Shuttle Mission Simulator. Units are lbf on the vertical axis, seconds on the horizontal.

enter image description here

Edit: Just noticed you asked about flowrates too. Again from SMS SSME model notes, here are simplified plots of the startup flowrates. Red is LO2, blue is LH2, Y axis units are lbm/sec.

enter image description here

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