Lot's of us, me included are interested in reading up on this fascinating halachic discussion.
For English Articles:
(1) See the aforementioned article by R’ JD Bleich published in Tradition (
http://traditionarchive.org/news/originals/Volume%2036/No.%203/Survey%20of%20Recent.pdf). See also the letter from Cyril Domb regarding R’ Goren opinion, and R’ Bleich’s response here:
http://traditionarchive.org/news/originals/Volume%2037/No.%202/Communications.pdf(2) Here is the article that most easily shows up in the world of google scholarship:
https://www.star-k.org/articles/kashrus-kurrents/515/when-does-one-pray-when-there-is-no-day/. For anyone interested, here is the more comprehsnive version that R’ Heber wrote in his Hebrew language Sefer:
http://hebrewbooks.org/pdfpager.aspx?req=53000&st=&pgnum=107(3) Another more simple read:
http://dinonline.org/2012/06/03/halacha-talk-the-exotic-traveler/(4) The following is not an orthodox source but quotes an array of sources, including the opinion of the Lubavitcher Rebbe.
http://www.schechter.edu/a-responsum-regarding-space-travel/. He cited the Lubavitcher Rebbe in “Yagdil Torah, New York, 1981, No. 73.” I have not been able to locate this source.
Hebrew Articles and Resources:
(1) R’ Boruch Simon source compilation on the topic:
https://www.yutorah.org/download.cfm?materialID=517617 (this includes the article from R’ Kasher, Minchas Elazar, R’ Posen. Another compilation of Hebrew sources:
http://beinenu.com/system/files/alonim/124_o.pdf?download=1 (2) R’ Taplin: תאריך ישראל עמוד תתמט (Can’t seem to find it online aside from Otzar Hachochma).
Between these sources, one will see addressed in regards to keeping shabbos:
(1) The preliminary question of whether the halakhic “day” requires a sunrise & sunset or a 24 hours (just about) period constitutes a day – this is largely based on proofs from Tanach, Gemaras, Midrashim (significantly a Pirkei D’Rav Elazar),
(2) Assuming a sunrise & sunset is required, is shabbos really every seven years!?!?,
(3) Assuming sunset & sunrise is not required to constitute a halakhic day (either on a d’orysa, or at least d’raban level [meaning, perhaps Shabbos on a d’orysa level is every seven years but to not “forget Shabbos” it should be kept on a rabbinic level every seven days – a similar, though not exact notion is noted when one is lost in a desert], what time does Shabbos begin or end:
(i) place where one left from [very ambiguous standard, what “place”],
(ii) lowest point of the sun is sunset, highest is sunrise (odd because lowest point is the sun’s general revolution (aka earth’s revolution) is chatzos, not sunset,
(iii) highest and lowest points of the sun are chatzos hayom and Layla, 12 hours in between each, six hours before highest point is sunrise, six hours later is sunset,
(iv) use the sunset/sunrise of the same longitude,
(v) Just use Israel zmanim (Jerusalem?).
Note A: These times help for actual sunset (shkia); how to calculate tzeis, if there is a tzeis, may be harder. If ppl like using the somewhat arbitrary 60 (?), slightly less arbitrary 72 (or 90 !?!) this may get more complicated as to what makes sense. If Tzeis is amount of degrees below the horizon, then it should be easy to figure out (relatively I guess) based on the sun’s descend from the mid-point from highest to lowest (after hour six) the generally assumed proper degrees.
Note B: These times are to figure out the halakhic day which relates most significantly to Shabbos. There are other questions (i.e. Krias Shema) which may require a separate analysis because these may require a halakhic “Layla/night” which would not exist when there is daylight all 24 hours (and vice versa in the winter) – see Rabbi Shternbuch in the quotes מועדים וזמנים who makes this salient point. Or perhaps because Shema is not a “night” based on mitzvah but a “sleeping/awaking” based on mitzah it will apply and be even less complicated than Shabbos question.
ותן לחכם ויחכם עוד.