Assuming a hotter day
Colder than the corner, then, there is something in hand. The engine is holding to the rated figure with its turbine running well short of what it could stand, and that unused margin is what the rest of this chapter is about.
Suppose today is well colder than the corner temperature and the runway is far longer than you need. You do not need full thrust to get airborne, so you would rather not use it — because thrust is what wears engines out, and it wears them out expensively, with the wear climbing steeply as the temperature climbs.
There is already a mechanism that gives less thrust for a hotter day — the temperature ceiling you have just watched at work, taking thrust away with every degree past the corner. So you tell the airplane the day is hotter than it is. You give it an assumed temperature somewhere warmer than the corner, and the engine sets the thrust it would have produced on that day — which is less than today's thrust, exactly as intended.
It is legal because it is checked: the reduced thrust must still get the airplane off the runway you actually have, meet every climb requirement, and clear every obstacle, at today's real weight. You are not pretending the takeoff is easier. You are pretending the engines are weaker, then proving the takeoff still works with weak engines.
And the crucial property: the thrust is still there. Nothing has been disabled. Push the levers up and you get everything the engine can give on the actual day — the full rated figure if the day is colder than the corner, and the day's own temperature-limited maximum if it is warmer. Either way it is exactly the thrust you would have had without the pretence. You simply chose not to ask for it. If the takeoff turns into something else — a failure, shear, an obstacle you did not expect — the levers go forward and the whole engine is available, immediately. That is why the technique is acceptable at all.
There are two limits on the temperature you may assume, and both follow from what flat rating does.
The floor is the flat-rate temperature itself. Assume anything colder than it and you have achieved nothing whatever, because colder than the corner the engine delivers its full rated thrust no matter what figure you hand it. Sitting on top of that is a second floor: the assumption has to be warmer than the day you are actually standing in. A lower one would be asking the engine for more thrust rather than less. Whichever of those two is the higher is the lowest temperature you may assume.
Derate: a different engine for the day
Derate looks similar from the flight deck but is a fundamentally different thing.
A derate is not a pretend temperature. It is a lower certified rating — for this takeoff the airplane is being operated as if it had a new set of engines, with its own set of published limits and its own performance data. And that has one consequence which the assumed-temperature method cannot offer.
Think about what happens when an engine fails. The thrust on the live side is what swings the nose, and the rudder must be able to hold it. The speed at which the rudder can just manage that is a control limit, and it depends directly on how much asymmetric thrust there is to fight.
With less thrust certified, there is less asymmetry to control — so the minimum control speed comes down. Which can matter enormously on a short or slippery runway. Remember that V1 can never be set below the ground minimum control speed — that was the floor, and it does not bend. So when the floor is what is holding V1 up, the only way to get a lower V1 is to lower the floor itself, and a derate is what lowers it. A lower V1 is exactly what a poor stopping surface wants.
An assumed temperature cannot do the same, and the reason is the property that made it attractive in the first place. The full rating stays available throughout, so the control speed has to be calculated on it, in case you use it. An assumed temperature buys no reduction in control speed at all.
The price is that a derate is a genuine limit. Unlike an assumed temperature, you cannot simply push through it: for this takeoff you have a new set of engines, and a new set of engines comes with a new set of speeds.
And the reason is not paperwork. The lower control speed came from the derated thrust, and V1 may well have been set below what the full rating would have demanded. Push the levers to the firewall after an engine has failed and the asymmetry comes back to a value the rudder cannot hold at the speed you are doing. The result is a loss of directional control.
All of which makes the derate look like the poorer bargain of the two, and raises a fair question. Why bother with it at all?
For one good reason, quite apart from the control speed. The two can be combined. A derate sets the certified basis, an assumed temperature is then taken on top of it, and the pair together buy a far deeper reduction in thrust than either would manage alone. Manufacturers differ on whether they allow the combination.
Why would anyone choose a derate over an assumed temperature?
For the minimum control speed. A derate lowers the certified thrust, so an engine failure produces less asymmetry, so the rudder can cope at a lower speed. That opens the door to a lower V1, which is precisely what you want when stopping is expensive — a short runway, or a wet one. Remember that V1 can't be lower than Vmc, so sometimes the only way to lower V1 is to lower Vmc with derate. An assumed temperature cannot do this, and the reason is the thing that makes it attractive elsewhere: the full rating can be recovered at any moment, so the control speed must be computed on the thrust you could take back rather than the thrust you set. The same recoverability is why a derate remains available in conditions where an assumed temperature does not.