There are 12 Trophies in this title.

Bronze

ROLL
Performed a roll in Arcade mode.

Climax Mode
Activated Climax Mode in Arcade mode.

50 Combo
Got a combo of 50 in Arcade mode.

Stealth bomber
Cleared Stage 5 in Arcade mode.

Across enemy lines
Cleared Stage 10 in Arcade mode.

Emergency Order
Cleared an Emergency Order in Arcade mode.

Rank In
Uploaded a Score Attack score to rankings.

Triple A
Get a AAA ranking in Arcade mode (for Shot Down, Combo, or Play Time).

Silver

Super Tomcat
Cleared Arcade mode with the F-14D.

Super Hornet
Cleared Arcade mode with the F/A-18E.

Strike Eagle
Cleared Arcade mode with the F-15E.

Gold

All Clear!
Cleared Arcade mode with ending A.

http://www.thesixthaxis.com/2010/04/21/afterburner-climax-trophies/