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I notice the term LNAV engagement being frequently mentioned in aviation related discussions or in several FAA's relevant material. Take, for example, this quote from Order 8260.58A (PBN Instrument Procedure Design) which states:

Order 8260.58A [5-1]: The ICA must be long enough to allow LNAV engagement (500 feet above airport elevation).

I am aware that LNAV means "Lateral NAVigation" but I have some trouble understanding what does "engagement" exactly mean and why an aircraft have to climb at 500 feet above airport's elevation before the initial turn for this engagement to happen. Could someone point me in the right direction?

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In that context, "engagement" refers to transfer of control of the airplane to the autopilot. Transport aircraft will generally have a limitation that forbids autopilot engagement on departure below, say, 500 or 600 ft. Once it's engaged in NAV mode, the A/P needs a bit of time to sort itself out and start turning to follow the LNAV guidance.

So the design guidance there says that the lateral space allotted for the initial climb area for obstacle clearance has to take into account the behaviour of the aircraft when A/P is engaged at the minimum altitude allowable, in terms of the potential lateral real estate consumed to get on the proper departure track.

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Some old aircraft (737 Classic comes to mind) don't allow arming LNAV/NAV AFDS mode on the ground, as you normally have to be at least 400ft above ground (ICAO PANS OPS says 396ft) before making turns more than 15 degrees, at which point you would engage LNAV. I guess the regulation takes old aircraft into account.

On the newer aircraft, you can arm LNAV/NAV mode on the ground already and it normally becomes active at 50ft.

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