James, so when the needle moves as you blow right throuhg the final approach course, you think you're "established" on final?
There's no technical regulatory definition. This time the FAA is using plain everyday English.
Under FAR 91.181, we are generally expected to fly the centerline of any course. In fact the Pilot/Controller Glossary defines "ON-COURSE" as
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"ON-COURSE - Used to indicate that an aircraft is established on the route centerline."
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Obviously, there are tolerances built into the system since neither people nor navigational aids are perfect.
But for "established", I think the FAA means it in the same sense as the plain English definition.
From the Pilot/Controller Glossary
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ESTABLISHED - To be stable or fixed on a route, route segment, altitude, heading, etc.
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And, from the American Heritage Dictionary of the English Language
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a. To place or settle in a secure position or condition...
b. To make firm or secure
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So, are you "stable" on a route when the needle moves off the peg and you begin to make your turn? Or are you "stable" when you complete your turn, the OBS and your flight path (roughly) agree, and the needle is no longer moving (except for normal bracketing corrections)? Or is it somewhere in between?