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The portion of the hours of darkness when satellites can be seen depends upon your Latitude and the season.
The Weather conditions need to be taken into account as well as the phase of the moon.
We can see stars at night as they all shine by the light of their own. The moon, the planets and satellites do not generate light but are lit by the sun instead. During the night on Earth, we are standing within the Earth’s own shadow and the sun is below the horizon. Stand on a ladder and you won’t expect to see the sun, walk to the top of a hill, still no sun, fly up in a plane to 4000 metres; nope! Fly up in a rocket to 300 km or so and aha! We see the sun above the horizon again. It’s now illuminating you but the ground down below is in darkness with city street lights, fires and flares from oil rigs.
At sunset there is a much better chance of experiencing this on a small scale if you feel athletic enough. At the back of my old school (Upbury Manor, Gillingham, Kent UK), there is a 40° inclined hill about 60 metres high that lead down to the town of Chatham. In November and February the sun sets around 4pm. I used to walk to the bottom of that hill after school on a very clear day and watch the sunset. Then I ran half way up as fast as I could, turned and watched the sunset again, then ran to the top and saw a third sunset. The bottom of the hill was getting dark while the top was still lit by the sun.
Now imagine a ladder, climb up it quick and witness a fourth sunset, then onto a small plane up at 1000 metres, then a jet at 15,000 metres and a satellite at 400 km. Anybody can witness the sun set many times on the same day if you keep going higher at a faster rate than the setting sun. The Sky TV satellite is only in darkness for 45 minutes or so per 24 hr orbit. Far enough from the Earth and it becomes visible 24 hrs a day, but down below that part of the Earth may be in darkness. The deep space probes experience the sun constantly.
Satellite Spotting & Operations Handbook