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The Wikipedia entry for TCAS (Traffic collision avoidance system) states for resolution advisories (RA):

"When an RA [Resolution Advisory] is issued, pilots are expected to respond immediately to the RA unless doing so would jeopardize the safe operation of the flight" (Source).

Is there a general rule of thumb as to what "immediately" translates? How many seconds (or distance?) remain from time of resolution advisory to closest point of approach? Although RAs are usually accompanied with traffic advisories, I assume there is still some response time already calculated in...

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It depends a bit on speed but your linked article has the answer

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Traffic advisories are provided 35-48 seconds out and a Resolution will be provided 20-30 seconds out. Since the resolution is coordinated you only need enough time to clear the other aircraft however the greater separation the safer it will be so its advised to respond immediately upon a resolution being provided. Immediately means as fast as possible.

The FAA's TCAS II outline document states that modeling was performed assuming the pilots responded within 2.5~5 seconds

In modeling aircraft response to RAs, the expectation is the pilot will begin the initial 0.25 g acceleration maneuver within five seconds to an achieved rate of 1500 fpm. Pilot response with 0.35 g acceleration to an achieved rate of 2500 fpm is expected within 2.5 seconds for subsequent RAs.

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