Day lengths

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Day lengths

Postby SpitValve » Fri Jun 22, 2007 1:36 am UTC

Today is the shortest day of the year!

This reminded me of some stuff:

The stars rise 4 minutes later each day (~24 hours/365).

For your latitude, what times of year does the sun rise at least 4 minutes earlier each day? What other latitudes is this possible for?

This would effectively mean the constellations are "moving backwards" (in some very very vague sense) because when they become visible they are in an "earlier" position than the day before.

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Postby Alisto » Fri Jun 22, 2007 1:44 am UTC

I was going to yell at you and say that today was the longest day of the year. Then I remembered that you're an Aussie.
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Postby djn » Fri Jun 22, 2007 2:02 am UTC

Alisto wrote:I was going to yell at you and say that today was the longest day of the year. Then I remembered that you're an Aussie.


Except that he isn't, in a way I suspect is fairly important to those it concerns.

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Postby Hawknc » Fri Jun 22, 2007 2:13 am UTC

Alisto wrote:I was going to yell at you and say that today was the longest day of the year. Then I remembered that you're an Aussie.

Damn Mexicans, can't even get their countries right...
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Postby Alisto » Fri Jun 22, 2007 2:34 am UTC

Damn it.

Though as far as I'm concerned, New Zealand is like Australia's younger, more attractive sister.
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Postby 3.14159265... » Fri Jun 22, 2007 2:52 am UTC

I don't get the whole Australia is a continent, country thing. I avoid thinking about it...

and oh, in Afghan/Persian culture, the longest night, is more important than the longest day.

It is customary for those who are smart (thus know about this) and rich (thus able) to go out the whole night, looking for people that need any kind of help and give them that (food, medical care, other vitals)
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Postby djn » Fri Jun 22, 2007 4:19 am UTC

Given that the longest night (on this hemisphere, at least) falls around december 25th, I wonder if there's any relation at all to the christmas gift-giving? (St Nicolas, and all that)

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Re: Day lengths

Postby gmalivuk » Fri Jun 22, 2007 3:49 pm UTC

SpitValve wrote:For your latitude, what times of year does the sun rise at least 4 minutes earlier each day? What other latitudes is this possible for?


Dunno the answer, not gonna figure it out. But I will say that it's not possible too close to the equator, because day length doesn't change enough there. And where it does happen, it's going to be near the equinoxes, because that's when day length changes most rapidly.

I will also say that it never happens for you, there in Wellington, where the sun never seems to rise or set more than two minutes before or after it did the previous day. Which means anyone closer to the equator than you (more than 59 degrees from a pole) will also never see that happen.
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Postby SpitValve » Sat Jun 23, 2007 12:27 am UTC

ok... assume the sunset times go sinusoidally, which seem pretty true if you're not in the tropics.

So sunset time T = T0 + A sin (wt)

where w is about 2pi/365 days and t is the time of year, in days.

dT/dt = -Aw cos (wt)

We want Aw > 4 minutes/day

Or A > 3.9 hours.

So you need the variation in time length to be something like 4 hours. That sounds fairly close to the poles...

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Re: Day lengths

Postby Cosmologicon » Sat Jun 23, 2007 12:43 am UTC

SpitValve wrote:The stars rise 4 minutes later each day (~24 hours/365).

For your latitude, what times of year does the sun rise at least 4 minutes earlier each day? What other latitudes is this possible for?

This would effectively mean the constellations are "moving backwards" (in some very very vague sense) because when they become visible they are in an "earlier" position than the day before.

It only happens in the Arctic and Antarctic Circles. I'm sure there must be some way to visualize this, but every time I try I get confused. So here's a derivation with calculus. First I derived the following:

cos(A) + tan(L) sin(B) tan(F) + sin(A) cos(B) tan(F) = 0

where A is the azimuthal angle (in the sidereal frame, the frame of the fixed stars) of the Earth's rotation at sunrise or sunset, L is your latitude, B is the Earth's axial tilt, and F is the azimuthal angle of the Earth's orbit, defined such that F = 0 is the vernal equinox. Implicit differentiation gives:

-sin(A) dA/dt + tan(L) sin(B) sec^2(F) dF/dt + cos(A) cos(B) tan(F) dA/dt + sin(A) cos(B) sec^2(F) dF/dt = 0

We want the smallest L for which dA/dt = -4 min/day at some point. (Since there are some imperfect assumptions already at this point, I won't be fussy about sidereal vs solar day.) gmalivuk is right: this happens on the equinox (F = 0 for the northern hemisphere), so tan(F) = 0 and sec(F) = 1. Also, at this time, the sidereal and solar frames match up, so A = 0 at noon, or -270deg at sunrise. Subbing those in:

dA/dt = -dF/dt (tan(L) sin(B) - cos(B))

A more useful value than A would be the sidereal time of sunrise T. This is related to A by a factor of 2pi/day. Meanwhile dF/dt is a constant at 2pi/year. Also, if we want it in solar time instead of sidereal time, we just need to sutract 1 from the right:

dT/dt = -day/year * (tan(L) sin(B) - cos(B) - 1)

dF/dt is a constant, 2pi/year, and B is just 23.44 deg, so it's easy to get dT/dt for any given latitude. For instance, for my location, L = 42.4 deg, and so dT/dt = -0.00122 = -1.76 min/day. According to USNO, this year sunrise was at 05:55 five days before the equinox and 05:38 five days after, for a difference of -17 minutes in 10 days. So that confirms it. Wellington would have a similar value during the autumnal equinox.

The value 4 min/day is exactly the same as day/year (as SpitValve said), so for dT/dt = -4min/day, we have tan(L) sin(B) - cos(B) = 0, so L = 90 deg - B, putting it on the Arctic Circle.

I find it interesting that at this same latitude, the sunrise is not getting 4 minutes later each day 6 months later. That occurs north of the latitude for which tan(L) = (2 - cos(B)) / sin(B), or L = 69.8 deg. I expected there to be symmetry here, but there's apparently not.


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Postby gmalivuk » Sat Jun 23, 2007 8:04 pm UTC

SpitValve wrote:ok... assume the sunset times go sinusoidally, which seem pretty true if you're not in the tropics.


I think the problem is that the rate of change never gets quite as high as it would for pure sinusoidal, since the effect of being between perihelion and apihelion during the equinoxes effectively cancels part of the trend. (In that, if you plot sunrise times, the bit between solstices is more linear than it would be for a true sinusoidal trend.)
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Postby Delbin » Sat Jun 23, 2007 8:23 pm UTC

djn wrote:Given that the longest night (on this hemisphere, at least) falls around december 25th, I wonder if there's any relation at all to the christmas gift-giving? (St Nicolas, and all that)


There's a very strong relation. Christiaity absorbed a winter solstace celebration, trees and all, to make Christmas.

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Postby gmalivuk » Sat Jun 23, 2007 8:45 pm UTC

Delbin wrote:
djn wrote:Given that the longest night (on this hemisphere, at least) falls around december 25th, I wonder if there's any relation at all to the christmas gift-giving? (St Nicolas, and all that)


There's a very strong relation. Christiaity absorbed a winter solstace celebration, trees and all, to make Christmas.


Only they fucked up, by about 4 days.
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Postby SpitValve » Sat Jun 23, 2007 9:01 pm UTC

gmalivuk wrote:There's a very strong relation. Christiaity absorbed a winter solstace celebration, trees and all, to make Christmas.


Only they fucked up, by about 4 days.[/quote]

I don't think getting the date exactly right was hugely big on their minds...

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Postby yy2bggggs » Sat Jun 23, 2007 10:00 pm UTC

gmalivuk wrote:Only they fucked up, by about 4 days.

That's an anachronism I believe. Compare the history of the belief that Jesus was born on the 25th of December (or the Saturnalia festival if you prefer) to the history of the calendar you're claiming they picked the wrong date from.

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Postby tessuraea » Sun Jun 24, 2007 12:15 am UTC

Yeah... people didn't really agree on calendars for a long time. Jesus was reputedly born on the birthday of Mithras, which was also Saturnalia, and coincided, more or less, with the winter solstice. Exact dates don't matter. And Jesus' birthday wasn't fixed at the winter solstice for a long time--I don't remember exactly when. There are plenty of indications in the Bible itself that it wasn't actually then. The best one I can remember is that there were lambs being born. Lambs are not born at the start of winter.

Winter solstice is a very auspicious time for a leader to be born--because the sun is being reborn then too, in the context of the year.
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Postby SpitValve » Sun Jun 24, 2007 12:38 am UTC

Did anyone actually believe he was born on the 25th of December? I always figured they just arbitrarily set it as a festival to celebrate his birth, because there's no way of guessing what his birthday was.

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Postby djn » Sun Jun 24, 2007 1:08 am UTC

SpitValve wrote:Did anyone actually believe he was born on the 25th of December? I always figured they just arbitrarily set it as a festival to celebrate his birth, because there's no way of guessing what his birthday was.


The idea I've got is that it's just date choosen for its general suitability. Then again, that doesn't mean people don't believe it; There's lots of scary people out there that will happily ignore evidence to the contrary.

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Postby Vaniver » Sun Jun 24, 2007 1:53 am UTC

Did anyone actually believe he was born on the 25th of December? I always figured they just arbitrarily set it as a festival to celebrate his birth, because there's no way of guessing what his birthday was.
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Postby Cosmologicon » Sun Jun 24, 2007 1:37 pm UTC

gmalivuk wrote:
SpitValve wrote:ok... assume the sunset times go sinusoidally, which seem pretty true if you're not in the tropics.


I think the problem is that the rate of change never gets quite as high as it would for pure sinusoidal, since the effect of being between perihelion and apihelion during the equinoxes effectively cancels part of the trend. (In that, if you plot sunrise times, the bit between solstices is more linear than it would be for a true sinusoidal trend.)

Even if you assume a perfectly spherical orbit, it's pretty far from sinusoidal, according to my calculations, which seems to match up with the calculated tables to first order.

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Postby NathanielK » Sun Jun 24, 2007 7:23 pm UTC

djn wrote:(Capital after semicolon or not?)
You shouldn't capitalize the next word after a semicolon; without a period, it's not actually a different sentence.
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Postby SpitValve » Mon Jun 25, 2007 5:00 am UTC

Cosmologicon wrote:Even if you assume a perfectly spherical orbit, it's pretty far from sinusoidal, according to my calculations, which seems to match up with the calculated tables to first order.


Kinda looks loosely sinusoidal...

But guessing equations from graphs is not hugely rigorous :)

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Postby Cosmologicon » Mon Jun 25, 2007 5:14 am UTC

Wow, it looks significantly non-sinusoidal to me! As the x-axis goes from 2 to 4, the y-axis goes from 6.9 to 5.1, for a slope of -0.9. As the x-axis goes from 8 to 10, the y-axis goes from 5.35 to 6.65, for a slope of +0.65. If it were sinusoidal then these would be equal in magnitude.

I mean, I guess it depends on how exacting you are, but it seems like a big difference to me. I guess it goes up and down, at least....

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Postby gmalivuk » Mon Jun 25, 2007 3:43 pm UTC

SpitValve wrote:
Cosmologicon wrote:Even if you assume a perfectly spherical orbit, it's pretty far from sinusoidal, according to my calculations, which seems to match up with the calculated tables to first order.


Kinda looks loosely sinusoidal...

But guessing equations from graphs is not hugely rigorous :)


Especially if the graphs are made from a sinusoidal approximation instead of actual data...

Also, 45º is far more sinusoidal than any of the parts where the actual change per day gets up to 4 minutes. But still, not quite:

Image

The variations in transit (local noon) were multiplied by 5 to get them to show up well at this scale, which is why that graph (very much not like the one in your picture) is so rough. But notice that even at 45ºN, the actual sunrise times are not the same as predicted by a sinusoidal map with the same average, period, and maximum.
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Postby SpitValve » Tue Jun 26, 2007 7:36 am UTC

I'd still say the sunrise times look close enough to sinusoidal that for e.g. a 1st year assignment, I'd let them assume it was so.

But it's quite interesting that the transit is so different to a sinusoid.

(and yeah, I did kinda suspect the graphs might not have been from real data, I tried to find one but couldn't...)

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Postby Cosmologicon » Tue Jun 26, 2007 4:28 pm UTC

The variation in transit time is definitely due to the eccentricity of the Earth's orbit, combined with the tilt of the Earth's axis. It's a well known effect called the Equation of Time. IIRC, it can cause the transit to vary from one day to the next by as much as 20 or 30 seconds. The discrepancy between the sunset variation and a sinusoidal approximation that I noticed is comparable magnitude: depending on what sinusoidal you compare it to, it's at least (1-cos(23.44 deg)) 4 min = 20 seconds.

I think a first-year could handle it, but it depends on what you expect of them, I guess. You don't want your textbook to be cited in a paper called "Common Misconceptions in Celestial Mechanics", do ya?

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Postby SpitValve » Wed Jun 27, 2007 7:24 am UTC

Cosmologicon wrote:I think a first-year could handle it, but it depends on what you expect of them, I guess. You don't want your textbook to be cited in a paper called "Common Misconceptions in Celestial Mechanics", do ya?


Good call :)


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