Of course, you can argue that it isn't mandatory in the United States. That is technically true but it is mandatory elsewhere in the world. I will let you decide your course of action from the sources below, including the G450 Aircraft Operating Manual, the 14 CFR series, and ICAO Doc 8168, Volume 1.
Me? I follow the RA and let ATC know what I've done.
Everything here is from the references shown below, with a few comments in an alternate color.
[G450 AOM, § 03-08-90]
Upon initiation of a Resolution Advisory (RA), the pilot flying should focus attention on flying the commanded maneuver. The airplane should be smoothly maneuvered to comply with the TCAS command and to promptly return to ATC clearance when the "CLEAR OF CONFLICT" message is received.
During a resolution advisory (RA) the HUD will present a small fly-to-box and provides vertical flight path guidance. Maneuver the flight path marker into this small box. On the PFD during a RA there will be a wider box and vertical pitch boresight symbol provided. Maneuver the pitch boresight symbol (single cue or cross pointer) into this wider box. For Airplanes with ASC 908 and prior, if the HUD cue is in use on the PFD, then it will switch to a single cue pitch reference automatically to provide the proper symbol. Airplanes with ASC 909 and subsequent will retain the HUD cue during an RA. There may also be a FPM displayed on the PFD if FPA is selected on the display controller or for airplanes having ASC 037, when SV-PFD is being used.
DURING A RESOLUTION ADVISORY (RA), CHOOSE EITHER THE HUD OR PFD AND MANEUVER THE AIRPLANE TO FLY THE APPROPRIATE SYMBOL INTO THE FLY-TO-BOX. IF USING THE HUD, MANEUVER THE FPV INTO THE SMALL FLY-TO-BOX. IF USING THE PFD MANEUVER THE BORESIGHT SYMBOL (SINGLE CUE OR CROSS POINTER) INTO THE WIDER FLY-TO-BOX.
Compliance with a TCAS II RA is necessary unless the pilot considers it unsafe to do so, or unless the pilot has better information about the cause of the RA and can maintain safe separation (e.g., visual acquisition of, and safe separation from, a nearby airplane, obvious TCAS system failure, etc.)
For airplanes having ASC 037 or ASC 037A (or later approved revision), with SV-PFD displayed the TCAS fly-to-box may flash due to “edge limiting”. If this occurs, continue to maneuver into the fly-to-box. It will command the proper escape maneuver.
I like to leave the autopilot engaged to preserve its thought process, including the cleared altitude and course. When the TCAS RA occurs, press and hold the touch control steering button and fly the airplane into the TCAS box. Once "clear of conflict," simply release the touch control steering button and the autopilot will return you to your previously cleared course and altitude.
[14 CFR 91, ¶91.123 (c)] Each pilot in command who, in an emergency, or in response to a traffic alert and collision avoidance system resolution advisory, deviates from an ATC clearance or instruction shall notify ATC of that deviation as soon as possible.
[AC 120-55C Air Carrier TCAS Appendix 5]
[49 CFR 830, ¶830.5] The operator of any civil aircraft, or any public aircraft not operated by the Armed Forces or an intelligence agency of the United States, or any foreign aircraft shall immediately, and by the most expeditious means available, notify the nearest National Transportation Safety Board (NTSB) office, when:
(10) Airborne Collision and Avoidance System (ACAS) resolution advisories issued when an aircraft is being operated on an instrument flight rules flight plan and compliance with the advisory is necessary to avert a substantial risk of collision between two or more aircraft.
So how do you notify the NTSB about a TCAS event? They have an email address set up to do just that: email@example.com.
[G450 AOM, § 2B-06-90]
[G450 AOM, § 2B-18-20 ¶2.D.]
Flying the flight path vector into the TCAS box in the HUD will keep you safe but it may not keep your right seat pilot happy. If you are climbing, placing the flight path vector in the top of the TCAS box will keep the right seat display green. If you are descending, placing the flight path vector in the bottom of the TCAS box will keep the right seat display green.
[14 CFR 91, §91.123]
(a) When an ATC clearance has been obtained, no pilot in command may deviate from that clearance unless an amended clearance is obtained, an emergency exists, or the deviation is in response to a traffic alert and collision avoidance system resolution advisory. However, except in Class A airspace, a pilot may cancel an IFR flight plan if the operation is being conducted in VFR weather conditions. When a pilot is uncertain of an ATC clearance, that pilot shall immediately request clarification from ATC.
(b) Except in an emergency, no person may operate an aircraft contrary to an ATC instruction in an area in which air traffic control is exercised.
(c) Each pilot in command who, in an emergency, or in response to a traffic alert and collision avoidance system resolution advisory, deviates from an ATC clearance or instruction shall notify ATC of that deviation as soon as possible.
(d) Each pilot in command who (though not deviating from a rule of this subpart) is given priority by ATC in an emergency, shall submit a detailed report of that emergency within 48 hours to the manager of that ATC facility, if requested by ATC.
(e) Unless otherwise authorized by ATC, no person operating an aircraft may operate that aircraft according to any clearance or instruction that has been issued to the pilot of another aircraft for radar air traffic control purposes.
The CFR does not require you follow the TCAS RA.
[AIM, ¶ 4-4-16 (as of Feb 9, 2012]
(a) The aircraft has returned to its assigned altitude and course.
(b) Alternate ATC instructions have been issued.
[ICAO Doc 8168, Volume 1, §3.2]
c) in the event of an RA, pilots shall:
1) respond immediately by following the RA as indicated, unless doing so would jeopardize the safety of the aeroplane;
Note 1.— Stall warning, wind shear, and ground proximity warning system alerts have precedence over ACAS.
Note 2.— Visually acquired traffic may not be the same traffic causing an RA. Visual perception of an encounter may be misleading, particularly at night.
2) follow the RA even if there is a conflict between the RA and an air traffic control (ATC) instruction to manoeuvre;
3) not manoeuvre in the opposite sense to an RA;
Note.— In the case of an ACAS-ACAS coordinated encounter, the RAs complement each other in order to reduce the potential for collision. Manoeuvres, or lack of manoeuvres, that result in vertical rates opposite to the sense of an RA could result in a collision with the threat aircraft.
4) as soon as possible, as permitted by flight crew workload, notify the appropriate ATC unit of the RA, including the direction of any deviation from the current ATC instruction or clearance;
Note.— Unless informed by the pilot, ATC does not know when ACAS issues RAs. It is possible for ATC to issue instructions that are unknowingly contrary to ACAS RA indications. Therefore, it is important that ATC be notified when an ATC instruction or clearance is not being followed because it conflicts with an RA.
5) promptly comply with any modified RAs;
6) limit the alterations of the flight path to the minimum extent necessary to comply with the RAs;
7) promptly return to the terms of the ATC instruction or clearance when the conflict is resolved; and
8) notify ATC when returning to the current clearance.
14 CFR 91, Title 14: Aeronautics and Space, General Operating and Flight Rules, Federal Aviation Administration, Department of Transportation
49 CFR 830, Title 49: Transportation, National Transportation Safety Board
Aeronautical Information Manual
Gulfstream G450 Aircraft Operating Manual, Revision 35, April 30, 2013.
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