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I have been wondering for a while, what is the differences between B.A.T.H, Full and Heading alignment, how long do each take and what are the pros and cons to each of them. 

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I have been wondering for a while, what is the differences between B.A.T.H, Full and Heading alignment, how long do each take and what are the pros and cons to each of them. 
The alignment process consists of two major phases: warmup and the actual alignment. Former depends mostly on outside temperature and varies from perhaps one minute to 20 minutes. Latter depends mostly on how stable your aircraft is (no movements allowed, not even bumping), but also on the actual position on earth. Overall, this can vary from a few to multiple to many minutes. I have seen alignments taking a hour because someone moved the aircraft just so slightly.

With stored heading, someone previously sat in your jet (e. g. ground crew), did a full alignment and then shut down the plane again without moving it from that position anymore. The previous alignment data (for that exact geo position) is now memorized in the aircraft and all you have to do now is the warmup phase. The actual alignment phase is skipped and the stored heading is taken instead.
Alignment quality is as good as it was beforehand. If you just spawned with it, its the same as a full alignment.

BATH and FULL differ in how precise they are. And hence also in the time it takes.
More on both in our manual:
https://f4.manuals.heatblur.se/systems/nav_com/ins.html#best-available-true-heading

From BATH you can expect roughly 5nm/h CEP and from FULL 3nm/h CEP. (in simple terms, the amount of average error in nm from the true position accumulated after one hour of normal flight)

In practice, rough numbers to orient yourself at are:
* STORED: 2min
* BATH: 4min
* FULL: 8min
(but as said, it may vary heavily if extreme conditions)

Worth noting that the warmup phase can be started with ground power already, before both engines are up. The actual alignment should be made only after to not cause issues or even damage when the power is shortly interrupted while moving from ground power over to the internal generators.
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