As discussed here, here, here, and here, there are different opinions as to when one is allowed to break their fast and end Shabbos, ranging from 36 minutes (mine) to 42 minutes to 60 minutes to 72 minutes to different times for fasts and Shabbos/Yom Kippur/Yom Tov. Where did these different opinions start? I understand that all of them are different calculations for when three stars come out, but why do different communities calculate that differently?
On a Torah level (mideoraysa), the appearance of three medium stars defines the onset of night: Shema may be recited from then onward and motzei shabbos occurs then. The most widely accepted views for when this occurs range from when the sun in 6.2 to 6.55 degrees below the horizon (though a view common in the US and UK requires waiting until this sun is 7.08 degrees below the horizon). This is not a set number of minutes, but varies according to latitude and season, among other factors.
However, for Krias Shema, the Shulchan Aruch, following Rabbeinu Yonah, requires waiting until three small stars have emerged. As R. Eliezer Melamed writes in his Peninei Halacha:
“The time to recite the evening Keriat Shema is “when you lie down” – when people lie on their beds to sleep, which starts when it gets dark. The Chachamim established a sign for the beginning of this time, when three medium-sized stars can be seen in the sky. This is because large stars are also visible during the day or at bein hashemashot (twilight). However, when three medium-size stars (according to the naked eye) emerge after them, it is a sign that night has begun (HaZemanim BaHalachah chapters 49-50). This time is called tzeit hakochavim. To avoid error, and to prevent mistaking big stars for medium ones, the Rishonim write that one must wait until he sees three small stars appear in the sky (Talmidei Rabbeinu Yonah; Shulchan Aruch 235:1).”
This is slightly earlier than the requirement of “three small stars in one place” for motzei Shabbos. Myzmanim does not give a time for “three small stars” (it uses 8.5 degrees as the default time for motzei shabbos/“three small stars in one place”), but R. Moshe Sokoloff reports that the person behind Myzmanim agreed that this should be around 8 degrees and that mirrors the ruling in contemporary works such as R. Dovid Borztyn’s Zmanim Kehilchasam. Others are comfortable with earlier times (either because they use an earlier time for three medium stars or because they believe there is less reason to wait for three small stars nowadays, when the potential for confusion has diminished due to our reliance on accurate clocks and calendars). As R. Eliezer Melamed writes, “However, so that people will not come to err, the Shulchan Aruch rules that they must wait until three small stars emerge, meaning, another few minutes. Some say that nowadays, since we use clocks, there is no need to wait. See HaZemanim BaHalachah 51, note 3.”
Some who hold this view sanction reciting shema from as early as 6.2 degrees onwards, though 6.45 degrees is more common (i.e. the time given as “end of ordained fasts according to R' Tukaccinsky” on Myzmanim).
As far as motzei shabbos times are concerned (where we wait for "three small stars in one place"), Dr. William Gewirtz writes, " 8.5 degrees is the most common current measurement for hashekhah, the requisite level of darkness15 that equates to the conclusion of Shabbat according to the ge’onim" and " With a few exceptions, the opinions of a vast majority of posekim equate to a point less than or equal to 8.5 degrees, with almost all falling in a range from approximately 7.5 to 9.3 degrees. Note that using 7.5 to 9.3 degrees for the conclusion of Shabbat, as opposed to approximately 6 degrees, parallels the extension from three medium stars to three small (and adjacent) stars."