Can I look at the sky and find the day of the week?
Suppose I wake up from a coma on a desert island in the 19th century (i.e. we already use the Gregorian calendar but have no satellites yet). I have a clear view of the sky and a couple of days to make observations and calculations. Can I determine the current day of the week? Also, can I find the answer instantly?
On a related note, is it possible to define days of the week in a purely astronomical (e.g. planet-agnostic) manner such that for the Earth we get the traditional meaning?
It is possible to determine the day of the year using the distance of the sun from the vernal equinox (you have to account for the equation of time etc, but it is possible). It is also possible to determine the year if your measurements are precise enough, using the current position of the vernal equinox (which changes due to precession). The ambiguity of the factor of 26000 years can be removed by noting the positions of other stars (for example, Barnard's star). Knowing these two, you should be able to find the day of the week.
About your second question, Astronomers like to deal with Julian date, which is a continuous day counting system, and you can determine the day of the week from the Julian date, though only if given a reference, for example Jan 1, 2000, was a Saturday. As such, a week and days of a week are man made (https://en.wikipedia.org/wiki/Week).
Ok, the first part is clear now, thank you, but I still don't get the second one. Perhaps I wasn't clear in the question. I accept that the days of the week are man-made, but that doesn't mean that they can't be absolute. For instance, the metre is an ad-hoc man-made unit of measurement, but today we define it as “the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second”. I wonder if there exists a similar definition for, say, Sunday.
By the way, I just realized that the Gregorian reformation didn't affect the days of the week at all, so we can use the Julian calendar as well.
About the absoluteness, you only need to define one day (Jan 1, 2000 you can say is defined as being a Saturday). This makes all other days absolutely defined. And afaik, there is no physical/astronomical definition for it apart from an arbitrary definition/assignment.
I see. I hoped for a bit more, but it seems like I'll have to face the reality. :)
In that case, the 'afaik' part should help. You (or someone else who's reading this) could research a bit more as to whether there actually is a real origin for at least some particular day being a particular day of the week. This is how you find unanswered questions and try to solve them as part of a research career. I would love to know if there actually is a specific origin or it's plain arbitrary. My knowledge is my limit (currently), not yours.
The internationally accepted week system has cultural and religious origins. Other than the fact that a 'day' is defined by an astronomical event, the week or its individual days have no significant relationship with astronomy. For this reason, you could calculate it using an index point of a day with a known day of the week as Takku has already described, but not without.
Ok, it's not the index point that really bothers me. I understand that it's a cultural and religious issue, but that only affects shifting and there still remains an astronomical question beyond that. Let me propose another thought experiment to make it clear. I'll drop you off at several planets and each time you should tell me which day of the week it is. You can say whatever you want, but if I take you back to the same planet at some later time, you'll have to be consistent (e.g. if you said Wednesday first and I bring you back 100 “days” later, you should say Friday now). Can you do that?
Given no other information, no. The method described relies heavily on knowledge of local bodies from which to derive the number of days since a point of index. This is very easy on Earth due to the abundance of information available to us. On a hypothetical planet in which this information isn't available, we have no way to derive days passed.
I agree with Mitch. We have defined references to an exact earth day & an exact earth year and we know about the precession & nutation periods in terms of earth years, from which we can get the date. Also, weeks are defined for earth days, so if we do go to other planets, we'll still have to work with earth days. We don't know the constants as accurately for other planets (and definitely not for hypothetical planets, but I don't think we're dealing with those here). And it is nearly impossible to calculate such relations in a time frame as small as a couple of days as was mentioned in the OP.
I've been thinking a bit, and yes, you're right. Suppose we have a star far enough from the others with a single planet orbiting around it and no other spatial bodies nearby. Then the planet has a perfect elliptical orbit. Now assume that a "year" consists of precisely 100 "days". If I drop you off on the planet twice with a difference of a "year", you'll see exactly the same (as the entire system is in the same state), but you need to give a different answer. For a deterministic method, that's a contradiction.