The second Gulf of Sidra incident, January 4, 1989, was when US F-14 Tomcats shot down two Libyan MiG-23 Flogger Es which appeared to be attempting to engage them, as had happened previously in the Gulf of Sidra incident (1981).
In 1973 Libya had claimed much of the Gulf of Sidra as its territorial waters and subsequently declared a "line of death", the crossing of which would invite a military response. As part of its ongoing freedom of navigation activities in support of 12 mile territorial waters practices the US Navy aircraft carrrier USS John F. Kennedy was operating near the Libyan coast.
At 20,000ft at 11:57AM on the morning of the 4th, VF-32 Gypsey Swordsmen F-14As AC207 (Joseph Bernard Connelly/CDR Steven Patrick Collins)(159610 AC207) and AC204 (Herman C. III Cook/Leo F. Enright)(159013 AC204) were flying combat air patrol about 70 miles from the Libyan coast. Two Libyan MiG-23 Floggers which had taken off from Al Bumbaw airfield were detected flying towards them. At the time the Floggers were 72 nautical miles away at 10,000ft and heading directly towards the Tomcats and carrier.
The F-14s turned away from the head on approach to indicate that they didn't want to fight. The Floggers jinked to return to head on approach at a closing speed of about 1,000mph. The F-14s dove to 3,000ft to give a clear radar picture against the sky and leave the Floggers with sea clutter to contend with. At 11:59 the radar-intercept officer (RIO) of the lead Tomcat armed the AIM-9 Sidewinder and AIM-7 Sparrow missiles it was carrying.
The Tomcats tried three more maneuvers to end the approach. Each time observers in an E-2C Hawkeye heard the Libyan ground controller instruct the Floggers to maneuver into a head on course.
At noon the trailing Tomcat locked on to the Floggers with its radar, which in past encounters had been reported to the Libyan ground controller and resulted in an instruction to break contact. The US aircraft did not hear those communications this time. At almost 12:01 the lead Tomcat pilot said that "Bogeys have jinked back at me again for the fifth time. They're on my nose now, inside of 20 miles", followed shortly by "Master arm on" as he armed his weapons. At a range of 14 miles he fired a Sparrow radar homing missile and reported "Fox 1. Fox 1." At ten miles range he fired another Sparrow. Both missed.
The Floggers accelerated and continued to approach. At six miles range the Tomcats split and the Floggers followed the wingman while the lead Tomcat circled to get a tail angle on the Floggers for a Sidewinder heat seeking missile shot. The wingman successfully engaged with a Sparrow and one of the US pilots broadcast "Good kill! Good kill!" The lead Tomcat closed on the final Flogger and at 1.5 miles the RIO fired a Sidewinder, which exploded in the tailpipe of the target. One crewman broadcast "Good kill!" and "Let's get out of here". The Libyan pilots were seen to successfully eject and parachuted into the sea. The Tomcats proceeded north to return to the carrier.
Subsequent examination of stil photography from the Tomcats indicated that the Floggers were armed with AA-7 Apex missiles. Depending on the model, this can be either semi-active radarhoming or infra-red (heat seeking) homing.
Identifications of the Tomcats vary. The narrative above used the details from Air Aces . Another source  identifies the wingman as AC202 rather than AC204. Both agree on AC207 as the lead. The Time story reports the RIO arming and firing. The accuracy of that claim is unknown - it may be that only the pilot can do so.