3 ms·

A difficult problem related to this: prove that, for any location in the earth, the longest night of the year is adjacent to the shortest day. This is infuria

A difficult problem related to this: prove that, for any location in the earth, the longest night of the year is adjacent to the shortest day. This is infuriatingly difficult (because the lenght of a consecutive day-night pair needs not be 24 hours), and I'm beginning to think that it may not even be true.

ben_w 2y ago
Solar day or legal day? The former trivially proven true, because the sum is constant. I bet the latter has a counter example somewhere where the summer/winter time transition means the day (or night) length changes more than the difference between then and the solstice. Even if not that, time zones and calendars can be changed by law, so there's been a few entirely absent days — one of my dad's stamp collection anecdotes was about post where the response was dated before the original message, and neither was incorrect, because one was Gregorian and the other Julian.
- Sharlin 2y ago
  The length of the solar day is not constant, for two primary reasons (non-circularity of Earth's orbit, and the obliquity of the ecliptic [1]). It varies from the length of the mean solar day by up to 30 seconds, accumulating a seasonal error of up to 15 minutes or so compared to wallclock time. [1] https://en.wikipedia.org/wiki/Solar_time#Apparent_solar_time https://en.wikipedia.org/wiki/Solar_time#Apparent_solar_time
  - jvanderbot 2y ago
    But it varies continuously on scales much smaller than seasonal changes. And seasonal changes are what affect day/night times. Without discrete jumps, you can actually consider a solar-noon to solar-noon time cycle as a day, and see that day + night must add up to a constant for any given cycle, therefore more night means less day.
    - Sharlin 2y ago
      That's true. It offsets both sunrise and sunset the same amount.
jvanderbot 2y ago
I'll try using solar noon as a fixed duration. We know that the duration between solar noons is the same every day (24hrs + a little bit). So let's slice that up into day and night. Consider the two solar noons that adjoin the maximum night duration. I claim one of those two noons happens during the shortest day. We have midnight -> Sunrise -> noon -> sunset -> midnight -> sunrise -> noon -> sunset. We know noon .. noon is fixed at 24hrs +a bit. Same with midnight..midnight. We know that midnight..noon is about 12 hours (+ half a bit). This is just orbital mechanics. We know sunset..midnight..sunrise is periodic over the year, with one maximum duration. Therefore we know that noon..midnight..noon contains a periodic amount of dark time with one maximum. Let that time be |D|. Let |L| be the "light time" either the preceeding or following midnight..noon..midnight period - whichever has less light. Since midnight..noon..midnight..noon has 36 hrs (ish), then |D|+|L| = 36 (ish), so maximum darkness must have a minimum light time.

ben_w 2y ago

Solar day or legal day? The former trivially proven true, because the sum is constant. I bet the latter has a counter example somewhere where the summer/winter time transition means the day (or night) length changes more than the difference between then and the solstice. Even if not that, time zones and calendars can be changed by law, so there's been a few entirely absent days — one of my dad's stamp collection anecdotes was about post where the response was dated before the original message, and neither was incorrect, because one was Gregorian and the other Julian.

Sharlin 2y ago
The length of the solar day is not constant, for two primary reasons (non-circularity of Earth's orbit, and the obliquity of the ecliptic [1]). It varies from the length of the mean solar day by up to 30 seconds, accumulating a seasonal error of up to 15 minutes or so compared to wallclock time. [1] https://en.wikipedia.org/wiki/Solar_time#Apparent_solar_time https://en.wikipedia.org/wiki/Solar_time#Apparent_solar_time
- jvanderbot 2y ago
  But it varies continuously on scales much smaller than seasonal changes. And seasonal changes are what affect day/night times. Without discrete jumps, you can actually consider a solar-noon to solar-noon time cycle as a day, and see that day + night must add up to a constant for any given cycle, therefore more night means less day.
  - Sharlin 2y ago
    That's true. It offsets both sunrise and sunset the same amount.

jvanderbot 2y ago

I'll try using solar noon as a fixed duration. We know that the duration between solar noons is the same every day (24hrs + a little bit). So let's slice that up into day and night. Consider the two solar noons that adjoin the maximum night duration. I claim one of those two noons happens during the shortest day. We have midnight -> Sunrise -> noon -> sunset -> midnight -> sunrise -> noon -> sunset. We know noon .. noon is fixed at 24hrs +a bit. Same with midnight..midnight. We know that midnight..noon is about 12 hours (+ half a bit). This is just orbital mechanics. We know sunset..midnight..sunrise is periodic over the year, with one maximum duration. Therefore we know that noon..midnight..noon contains a periodic amount of dark time with one maximum. Let that time be |D|. Let |L| be the "light time" either the preceeding or following midnight..noon..midnight period - whichever has less light. Since midnight..noon..midnight..noon has 36 hrs (ish), then |D|+|L| = 36 (ish), so maximum darkness must have a minimum light time.