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Judaism in Space Part 1: Keeping Jewish Time
Are mitzvot and hence halakhot obligatory if one is off-planet? In the 5730 issue of the journal, No’am, Rabbi Ben-Zion Firrer claims that mitzvot can only exist on the earth. His evidence is a discussion in the Talmud on Deuteronomy 12:1 which reads, “These are the decrees and laws you must be careful to follow in the land that the LORD, the God of your ancestors, has given you to possess—as long as you live in the land.” The rabbis argue whether one is obligated to do the mitzvot both in the land and in the Diaspora. Rabbi Firrer maintains that “in the land” or “upon the earth” means one must be on the planet Earth to be obligated by mitzvot. Thus, anyone living off-planet are not obligated to observe mitzvot.
Rabbi Shlomo Goren claimed that mitzvot and halakhot are dependent on a twenty-four-hour cycle which does not exist off-planet. However, Goren says that humans should be obligated to perform such mitzvot off-planet regardless because there are no natural conditions off-planet in which a human could survive. Humans can only survive in artificial atmosphere composed of “elements transported from Earth.” Thus, ExoJews are technically still living on Earth since they are still entirely dependent on it.
Yet Firrer counters this argument by citing the Talmud which states that, since the bodies of water in the Land of Israel are not part of the land itself, a ship carrying supplies to the Land of Israel is not yet within its borders. The moment the bottom of the ship scrapes the land, it is officially within the borders and the sailors are obligated by mitzvot observed in the Land of Israel.
Similarly, Firrer argues, any Terran ship that comes in contact with another celestial body would be subject to the rules of that body. Anyone aboard the ship is no longer obligated to observe Terran halakhah. In the same issue of No’am, Rabbi Menachem Kasher refutes Firrer’s argument, maintaining that halakhic obligations are personal ones which follow Jews anywhere they live.
Lost Traveler

(א) דין ההולך במדבר בשבת ובו ב' סעיפים:
ההולך במדבר ואינו יודע מתי הוא שבת מונה שבעה ימים מיום שנתן אל לבו שכחתו ומקדש השביעי בקידוש והבדלה ואם יש לו ממה להתפרנס אסור לו לעשות מלאכה כלל עד שיכלה מה שיש לו ואז יעשה מלאכה בכל יום אפילו ביום שמקדש בו כדי פרנסתו מצומצמת ומותר לילך בו בכל יום אפי' ביום שמקדש בו:

(1) The Law of Someone Who Is Wandering in the Desert on Shabbat, 2 Seifim: 1. One who is wandering in the desert and does not when is Shabbat, counts seven days from the day he realized he did not know, and sanctifies the seventh day with Kiddush and Havdalah. If one has sustenance, one is forbidden from doing any work whatsoever until he uses what he has, and thereafter one should do work each day, even on the day he sanctifies, to achieve minimal sustenance. One is permitted to walk every day, even on the day he sanctifies.

The issue here is whether someone who settles off-planet will have to be fixed in this pseudo-Shabbat state, since it could be difficult to determine what time/date it is for them.
Hourly Problem
Rav Ada bar Ahavah’s opinion was that the year was specifically 365 days, five hours, 997 chalakim (minutes), and 48 regaim (seconds), which came to an extra 55 minutes and 25.4 seconds.
-Rav Ada bar Ahavah’s opinion was contained in a collection of baraitot that was known to the Rishonim; Meiselman, Torah, Chazal & Science, 67.
Herein lie two major issues for ExoJews: the first issue is that according to this ruling, they would need to exist in a perpetual state of Shabbat, so they don’t desecrate the actual Shabbat; the second issue is that if they needed to count twenty-four-hour days to calculate their own Shabbat, they may fall out-of-sync with their neighbors. For example, if Martian colonists exist on a twenty-four-hour-and-thirty-nine-minute day while Martian Jews exist on a twenty four-day, their schedule will slip. Shabbat would eventually begin in the middle of the night or in the morning. Let us imagine Shabbat began at two in the morning, and an ExoJew wanted to mark the beginning. They would need to wake up, make kiddush, say Shabbat blessings, then immediately make Havdalah.
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See Mishnah Berurah 344:3. Cf., however, Kaf ha-Ḥayyim 344:5 who cites authorities who maintain that the weekday payer should be recited even on the day observed as Shabbat. Kaf ha-Ḥayyim himself rules that the Sabbath prayer should be recited but that the musaf prayer should be omitted. Thus, the prayers on the day that is designated to be Shabbat is a hybrid of Shabbat and weekday.
Jewish Time in Terran Polar Regions
“It is necessary to reflect upon [the manner in which] those who dwell or travel in the lands near the poles should conduct themselves. For, in proportion to proximity [to the pole], the day becomes lengthened. There [are places in which] a month or two months and even longer may be a day to the extent that there exists a place where the day is prolonged to half a year and similarly the night is half a year. And under the pole there is no day and night at all; rather there is twilight [during] the entire year for in that place there is no sunrise or sunset because the [celestial] equator is its horizon. If so, how should they establish Shabbat there?” - R. Jacob Emden, Mor U-Kezi’ah 344
“The issue of Sabbath observance aboard a space ship is a novel extension of the much older question of Sabbath observance in the polar regions and adjacent areas in which daylight and darkness extend for months at a time rather than alternating in periods of approximately twenty-four hours. Determination of the prescribed time for morning, afternoon and evening prayers as well as for other time-bound mizvot presents the identical problem. Jewish commercial travelers reached areas relatively close to the North Pole long before anyone, Jew or gentile, seriously dreamed of space travel. - Bleich, Contemporary Halakhic Problems, 77.