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Here, atmospheric pressure/composition, temperature, and gravity won't be considered; let's assume that all of those are the same as on Earth.

Would we "rest/stay awake" however long or short it takes to rotate? Would we schedule business depending on that? Will our bodies adapt to that or will we break down because of too much or too little work or rest?

Would we stipulate a 24-hour period even if it's very out-of-sync with the rotation of that planet? This becomes very tricky if you've got multiple "sunrises" and/or "sunsets" within that 24-hour period, and those could happen on dramatically different points from one 24-our period to the next. It would be easier, though, if the rotation of that planet is very slow.

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    $\begingroup$ People who live in the far north adapt fine to 24 hours of daylight and night, so it wouldn't be a big deal. $\endgroup$ – Oldcat Oct 8 '15 at 23:57
  • $\begingroup$ Yes, but they still have 24 hour "days". If we had 30 hour days, for example, how would that impact humans who are accustomed to 24 hour days? $\endgroup$ – Frecklefoot Oct 9 '15 at 14:09
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    $\begingroup$ Related: Would a human individual adapt to a change in the day-night cycle? $\endgroup$ – DaaaahWhoosh Oct 9 '15 at 14:20
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    $\begingroup$ I think Oldcat is right that if it was a very long time, people would just wake up and go to sleep at scheduled times. Related, there were some studies on how long people would stay up if not influenced by clocks or the sun and the general conclusion was that people will live longer days. en.wikipedia.org/wiki/Michel_Siffre and nytimes.com/1989/05/24/us/… . Granted, this isn't as long as you're asking about but several people adjusted comfortably to 48 hour days. $\endgroup$ – userLTK Oct 10 '15 at 9:31
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    $\begingroup$ Some of us don't handle the normal dinural period. People with delayed sleep phase disorder will colonise Mars! $\endgroup$ – JDługosz Oct 10 '15 at 10:51
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If the difference is only small, we could probably adapt, after all some scientists and engineers at Nasa work on Mars time. But even with only a small difference (one sol on mars is only 40min longer than a day on Earth) there are problems with jet lag.

If the difference became greater we couldn't quickly adapt. It would be hard to adapt to a two hour day length difference, and just as the ISS astronauts do, we would stick to something similar to Earth time. The ISS uses UTC.

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As all others species the human being can only adapt to very slow changes in the important parameters of his environnement. Therefore it only depends on how fast does the rotation change and how deep the rotation is transformed. You could easily imagine that our biology is made for living on earth with the basic parameters, that rule our world since ages.

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