The minor fasts last from daybreak (alot hashachar) to the emergence of the stars (tzait ha-kochavim). Alot hashachar is when the first light begins to appear in the east. Tzait ha-kochavim is when three medium-sized stars are visible in the sky. There are different opinions as to when exactly alot hashachar occurs – either when the first light begins to appear in the east (when the sun is 17.5 degrees below the horizon) or a short time later, when the eastern sky is illuminated (when the sun is 16.1 degrees below the horizon).
There are also two major opinions regarding tzait hakochavim. It occurs either when experts and those with excellent eyesight can see three stars (when the sun is 4.8 degrees below the horizon) or when regular people can see three stars (when the sun is 6.2 degrees below the horizon).
There is a common mistake that people make regarding this issue. They think that there is a set interval between alot hashachar and sunrise and between sunset and tzait hakochavim, which is not true. Rather, these intervals depend on the time of year and the place. Therefore, one should use a precise calendar.
According to the letter of the law, we should follow the more lenient opinion, because these fasts are Rabbinic enactments. However, it is best to act strictly. Since we are already fasting all day long, it is preferable to add a few extra minutes in order to fulfill our obligation according to all opinions.
When the Tenth of Tevet falls out on a Friday, one must fast until after tzait hakochavim, even though Shabbat has already begun by then (SA 249:4).
One who flies from America to Israel will fast less time, because he is flying in the opposite direction of the sun. Every hour in flight shortens the fast by more than half an hour. If he flies from Israel to America, he will fast longer, because he is flying in the same direction as the sun. Every hour of flying adds more than half an hour to his fast. The rule is that the fast starts at alot hashachar and ends at tzait hakochavim, according to the place in which one finds himself at that time (Iggrot Moshe, OC 3:96).