Air carrier Captain reported a performing a go-around as an aircraft was taking off on the runway while the flight crew was on final approach. ATC stated there was proper spacing; but the flight crew believed that with the poor visibility and weather; it was unsafe to continue with the approach.

Narrative

Aircraft X was from ZZZ to AUS. AUS weather OVC012 3 SM -RA. Weird weather with 45 - 50 tailwinds until 800 ft. on approach and huge temperature inversion where temps at 2000 ft. were 17°C and at 500 ft. were 6°C. At 1000 ft. we were still fighting tailwinds and calling landing checklist complete; while breaking out of the clouds with limited visibility. Runway lights were then in sight. At 500 ft. stable" call was completed; at which point I focused on the runway and at 400 ft. stated that I believed an aircraft was on the runway. The FO (First Officer) believed it was the runway approach lights. At 300 ft. I stated again that it was indeed an aircraft. He called the Tower and told them we had an aircraft on the approach end of the runway. Tower told us they were a midfield departure. As Tower was stating this the aircraft was noticeably starting its departure roll. At 200 ft. I called and executed a go-around. At that point I suspect the aircraft was about 3000 - 4000 ft. down the runway; but my focus was on flying the aircraft and executing procedures for the go-around. The go-around was flown and executed well. We were vectored around for a second approach that was uneventful; except for the significant abnormal weather - tailwinds and temps on final. After landing; Ground asked us to call TRACON to inform him of what triggered our go-around. After parking; I discussed everything that I've stated here with TRACON. He told me that Tower Controller stated we had approximately 6100 ft. of spacing and the minimum allowed is 6000 ft. I do not believe that we had that spacing. With the poor visibility and abnormal weather; the very close spacing felt unsafe and uncomfortable. During the debrief; we both agreed that the go-around was a good safe conservative call and were both concerned about the tight landing to departure spacing.I feel that the significant tailwinds on final were the causal factor. The 45- - 50-kt. tailwinds would have accelerated our ground speeds and caused problems for ATC when trying to time aircraft arrivals; especially when sequencing them with departures. Likely they cleared the departure aircraft to takeoff when we were about 4-mile final; which usually might be sufficient. If we carried that huge tailwind; our ground speed would've been 45 - 50 kt. faster than usual and would've compressed the timing. Furthermore; the departure aircraft likely was being diligent in their operations and not rushing things as it was nighttime and the weather was rather poor."