I'd recommend showing this + the time zone (e.g. "browser local time, UTC+2"). My guess was that it's the local time of the location entered, which can matter when telling someone "hey, you'll be able to see a cool satellite looking out your northward window at 9 pm!". If you're worried about keeping a clean look, at least add a tooltip.

(I've also been trained to distrust any time that doesn't specify a time zone, because I've seen sites that default to some random US timezone, use your local time, use UTC, ...)

> I've also been trained to distrust any time that doesn't specify a time zone

My pet peeve is status pages that only show times in some US timezone. I'm supposed to know what the conversion from that timezone to my timezone is, and then the timezone to my servers (which are in UTC)...