A hypothetical question, but perhaps nonetheless illustrative of underlying principles : If by some method, humanity [or some non-human force] managed to stop the rotation of the Earth around its axis, would Shabbos still need to be kept?
Suppose the rotation was stopped on a Tuesday (GMT). Would we view days as progressing based on clocks, or would we view the day as forever standing still based on the Sun never setting?
How do we observe time-bound mitzvot where time and seasons are different?
On Earth, The problem dates back to the 18th century, when Jews started moving north, where daytime or nighttime can last for days or weeks. The Talmud is our guide:
Rav Huna says: If a man is wandering in the desert and does not know when Shabbat is, he should count six days [as weekdays] and keep one day as Shabbat. [Shabbat 69b] Rava says…, "Every day he may do whatever he needs to survive, even on Shabbat."
The final law says:
A wanderer who lost track of time must keep six "weekdays" followed by one "Shabbat", but he may not do anything forbidden on Shabbat, on any day, except to survive. He must act out of concern that the real Shabbat may be on ANY day. [Shulchan Arukh, Orakh Hayyim 344]
That same logic was used to observe two holy days in the Diaspora, compared to only one in Israel.
When day or night lasts for six months, such as near the poles, there are several opinions:
-18th-century rabbi Jacob Emden said: Count six days of 24 hours and keep the seventh as Shabbat.
-The Tiferet Yisrael [Mishnayot Yachin U’Boaz - Brachot: End Chap 1,(1782-1860)] said: Use the times for prayers of the place from where you came.
-The 19th-century Ben Ish Chai [Teshuvot Rav Pa’alim - Sod Yesharim 2:4, Sephardic, 1832-1909] said: Consider 6am to be sunrise and 6pm sunset.
-The Moadim U’Zmanim [Chalek Bais (2) Siman 155 in the glosses]: said:
-In summer, when the sun does not set, consider that a day begins and ends when the sun is at its lowest point in the sky, usually around midnight.
-In winter, when the sun does not rise, consider that a day begins when the sun is closest to the horizon, usually around noon.
In space, there are two opinions:
-Rabbi Ben Tzion Firrer [5730 issue of Noam]: argues that mitzvot are only applicable on earth, because the Torah says:
These are the statutes and judgments, which you shall take care to do, in the land [ba-aretz], which the Lord, God of your Fathers, gives you to possess all the days that you live upon the earth [ha-adamah]. [Deuteronomy 12:1]
But note that the Torah does not say: Don't do mitzvot outside the Land of Israel or the Earth.
-Rabbi Menahem Kasher [5730 issue of Noam] argues that mitzvot are incumbent in every environment. The themes of the festivals, of Shabbat remembering creation, of the daily prayers, are always relevant. So one must apply the same rules on the moon and in space as for the North Pole.
The second opinion is most likely to prevail. Possible details are:
-When orbiting the earth, use only the time measured at the place from which you left the earth. Others are more lenient and say: Keep Shabbat anytime it is Shabbat anywhere on Earth.
-When going far from the earth, use the clock on the wall of the spaceship, synchronized at liftoff with the time and place from which you left the earth, and follow the Jewish calendar for that place after that. This makes particular sense because observance can’t ever be truly simultaneous with the place of origin, because of the relativistic twin effect: One twin stays on earth and the other travels in space, and when the traveler comes back he is younger than his twin, and he has actually experienced less time.
-On Earth, Jews must pray towards Jerusalem. So, in space, Jews must pray towards the earth.
-Needless to say, one may do essential ship maintenance on Shabbat and holidays, for pikuach nefesh, to save lives.
On another planet, one might have to pro-rate the length of the day or the year. Rabbi Azriel Rosenfeld says: On Mars, a "day" is 24 hours and 39 minutes in earth time and a "year" is 687 days in earth time, so one must modify observance of the calendar accordingly. Also, when on the moon, one need not bless the full moon, a custom called Kiddush Levana, done at night outside between Rosh Hodesh and time of full moon. The Lubavitcher Rebbe concurred.
The Gemorrah rules that Shabbos comes by itself and it not dependent on a Beis Din, contrary to Moadim which are dependent on Kiddush HaChodesh כדאיתא בב"ב קכ"א.
I understand that "Beit Din" in the Gemorrah means our acting as a result of astronomical phenomena. For example, Yom Tov can only come a certain number of days after the Beis Din has ascertained that there was indeed a New Moon. This applies even nowadays, according to the Gemorrah in Rosh HaShannah. What the Gemorrah says about Yom Tov and Beis Din would be analogous to our acting due to a certain number of rotations of the earth. The Gemorrah means that Shabbos comes independant of astronomical phenomena. Therefore the answer that Shabbat comes independent of astronomical phenomena is valid.
Answer #2: Also, even if your question were at least a ספק, it isn't worse than what the Shulchan Aruch in Siman שדמ deals with (what to do if you find yourself in the desert and you aren't sure what day Shabbos is). You still have to keep Shabbos m'Chumrah. In your case, you know for sure when the last Shabbos was.
