When living in Stockholm, I came to appreciate the various levels of twilight and darkness, rather than thinking of day and night so strictly. The sun being low on the horizon also scatters light across the sky in ways that are very beautiful and last much longer than sunrise and sunset in Australia where I grew up.
The whole "6am sunrise and 6pm sunset every day of the year" thing at the equator is kind of mind blowing.
Another maybe counterintuitive fact is that (to a reasonable approximation) everywhere on earth gets the same number of hours of daylight over the course of a year.
There is a handy rule of thumb called the "rule of 12ths", used in seamanship / ocean navigation / tidal calculations (maybe it is used elsewhere too, this just happens to be where I recognize it from). I think it can apply to solar, seasons, etc. -- well, anything sinusoidally cyclical -- as a useful mental model:
If you divide half the phase of a cycle (peak to trough) into 6 hours duration or whatever appropriate unit, like 6 months, i.e. x-axis --
then going down from the top of the peak (or up from trough), the amount of y-axis change in each unit/hour is:
Hour (or month #): amount of change vs. peak-trough total (i.e. total = 2*A)
1: 1/12
2: 2/12
3: 3/12
4: 3/12
5: 2/12
6: 1/12
For us, the peak / trough are: June 21 to December 21, and the x-axis is 1 month units. And assuming maybe a 2 hour peak-to-trough difference in daylight time y-axis (depends on latitude you live of course), then each 1/12th equals 10 minutes.
So these days (late March) we are in the middle of the fastest decrease part, and each month we gain 30 minutes of daylight. Or, each day we are seeing sunset get pushed by like 1 minute.
see: https://en.wikipedia.org/wiki/Rule_of_twelfths, the diagram explains it better of course
"One of the more interesting features I hadn’t appreciated before is that when you get close to the Arctic circle, the length of the days is essentially a zigzag, straight up from the winter solstice all the way to the summer solstice and back down again."
I had noticed this too and wondered if it was exactly true, with the "zigzag" being straight lines - I thought there might be a simple proof of this fact based on some trig identities. There's not, because it's not true - the lines aren't exactly straight, even if you ignore solar refraction - but it's a very good approximation.
The calculated daylight even downplays the actual light at the high (or low) latitudes quite a bit. E.g. at latitude 60 there's a "nominal" midsummer night of about four hours, but it doesn't really get dark, as the light from the refraction is quite strong even with the disk not being visible.
I'm from Iceland, so a latitude of about 64.15 or so. The extremities of the graphs kind of describe how the mood of the people swings up and down all year round. In the middle of summer, around the summer solstice, people are bordering on mania, enjoying endless daylight and trying to get the most out of each day. As winter solstice approaches, everything becomes more subdued, a bit depressed even. It's often a bit difficult to live with, especially during the shortest winter days, but the summers are so incredibly amazing that it feels like it's all worth it.
The seasonal extremes of daylight are so extreme up here in Finland that the cycle of night & day seems a bit less like a 24-hour cycle and a bit more like a 365-day cycle.
An artifact of this is that my 5yo might not see a dark sky for the entire summer, unless we keep him up awake for the traditional Midsummer hangin'-out.
Coming from middle of Sweden I remember the first time I spent a midsummers night in Lund in the south if Sweden and was astonished that the night was in fact dark! In my hometown, well below the arctic circle, the month of June is still constant daylight.
I've been using my iPhone in standby mode at night and I have the world map with the daylight "sine wave" on. Its been fun watching the shape of the daylight region change as it get closer and closer to the top of the map. After the equinox it pops through the top of the map and then it's upside-down for the next 6 months.
Watching the changing shape of the daylight illuminated portion of the map over the year has given me a much better understanding of what the equinox means, why daylight hours shift, and just in general a better appreciation of our place in the solar system.
DST really bothers me. I actually have a wall clock set to local solar time, so that I can quickly see what time it "really" is. For some reason I've always been very sensitive to when sunrise and sunset is, and having them offset makes my brain feel weird.
An interesting analysis, covers a lot of ground, but no mention of the "analemma" (https://en.wikipedia.org/wiki/Analemma), an annual figure-eight curve that results from earth's elliptical orbit around the sun, and that must be included for reliable calculation of solar position and rise-set times.
Calculating the analemma requires multi-term Fourier analysis to reliably pin down the sun's position, without which nautical sextant calculations would fail. Even in the GPS era, professional sailors are required to (demonstrate their ability to) turn sextant sightings into geographical positions, against the possibility of a high-tech failure.
Apropos, during my around-the-world solo sail (https://arachnoid.com/sailbook), because of just such an equipment failure, I was obliged to navigate from French Polynesia to Fiji using a sextant -- and reliable solar positions.
I live just a couple hundred miles south of the arctic circle, and personally I hate the time of year where we "accelerate" into the equinoxes (equinoxii?). The rate of change is just too fast and too disruptive, and you _really_ see its effects on people. And then DST comes in and makes it even worse.
The difference as you climb in latitude is really shocking. Even just another 3-400 miles south of here, the rate of change is way less severe.
Anyway nice work and cool article! I've done some of these rough calculations myself before to plot out the change just to verify that I'm not insane for hating this time of year, and you did a way better job than I ever did :)
Something that is still crazy to me, regarding time zones, is that my hometown (where my mom lives and we visit frequently) is about 1.5 hours to the west, one timezone over.
When I’m taking to her on the phone in the evening, we’ll be talking about it getting darker.
The difference in the sunset 1.5 hours away in another timezone:
About 9 minutes.
This equates to my evening daylight lasting 51 minutes than hers, and her daylight in the mornings starting 51 minutes before mine (in our respective time zones).
Which seems to negate the whole effects of daylight saving time for one of us.
Of course this is exacerbated even more if you’re across the border of a timezone from each other (which I pretty much live on the western border of ET).
During a long hard winter night while I was longing for sun but the only warmth 1 got was from my screens, I coded
to give me data and hope.
A closely related follow-up blog post could be titled: “How Long Twilight Lasts”.
I have a wake-alarm[0] that triggers 30 minutes before civil twilight, that is roughly 60 minutes before local sunrise.
In the northern hemisphere at 52 degrees it gets earlier by about 2 minutes each day (additional 4 minutes of daytime).
So I get more sleep and short days in winter and less sleep and longer days in summer. It's liberating basing schedule on it and not some arbitrary time.
[0] https://f-droid.org/packages/com.forrestguice.suntimeswidget...
Nice iambic pentameter in that title.
How fast the days are getting long and so
The summer rains are melting all the snowThis phenomenon became very interesting to me after moving approximately 14 degrees further north (on the northern hemisphere) and experiencing not just shorter and longer days, but more rapid changes in day length during spring/fall.
The impact this has on daily life is larger than I had anticipated, and in general reducing the intensity of the cycle is a selling point for cities closer to the equator. It’s been nine years since my migration north, and I’ve only moved further north so this isn’t a deal breaker for me. It’s mostly something from my childhood and young adulthood that I took for granted. I’m now eagerly awaiting the day when my normal waking time is during dawn, which should be in early April.
I’m still significantly further south than Northern European countries mentioned in this thread. Maybe life has more moves north for me in store.
Very interesting discussion! I did a less theoretical approach a while ago to calculate the same thing: I just go to timeanddate.com, find the city I'm interested in, go to its sun page (example for New York https://www.timeanddate.com/sun/usa/new-york), and find the table that shows the difference in daylight minutes per day. No math or programming needed, just copy-paste and some buttons in Excel.
Not as satisfying as a derivation here, but a quicker way to get the answer.
EDIT: I did a spot check for Rovaniemi, Finland. This city is far north enough that the sun is up all day (66.5 degrees). But the graph on this page seems to be a little bit off: it requires an even higher latitude for that to show up.
Moving to Northern Canada has made me really really appreciate Spring. Going to work in the dark then going home in the dark is exhausting. I'm at 56ÂşN. I imagine it's worse for those who are even further North.
Do the maths for the southern hemisphere and compare: It is my belief due to axial tilt AND the solar rotation period and solar distance, the effects aren't entirely equal north and south pole.
Do the maths for the dusk side of the story. Dawn is good, but how this behaves in the rate of sun appearance and rate of sun disappearance also has a shift, the two event peaks aren't aligned. Again, a function (I believe) of the axial tilt.
So its a system with differences, with multiple inputs, with complex maths. I love it!
Fasting the month of Ramadan when it lands in March or September is always the most interesting for this reason. You have to change how you plan your daily routine as the month goes.
This is surprising. I had always assumed that the length-of-day function is essentially a sine wave everywhere on the planet, and that the derivative would thus be another sine wave shifted by 90 degrees.
When the day length is maximal/minimal (solstice), the day length change rate is near zero, and vice versa. That's still true in the more accurate model, even though the shape of the functions is more distorted.
The author, Joe antognini, also has an excellent podcast on the history of astronomy called: The Song of Urania
What am I missing here? It states:
> On the equator, every day of the year is exactly 12 hours long
But surely that is not true, as the Earth's rotation is tilted. Is this supposed to be a year round average? Despite calling out more advanced corrections one can make, this doesn't seem to be mentioned anywhere?
The author has a great podcast on the History of Astronomy
The Song on Urania https://songofurania.com/
It's taking a break for 2025 but there are enough episodes there to give you enough to listen to until he restarts
The graph gets crazy if you move the slider to be 66+ latitude.
Is the graph correct at those extremes? Like North Pole?
Another way to visualize this would be to look at the DOE NREL solar intensity maps for mid-winter vs. mid summer:
Can someone do a similar analysis which explains clearly why I can see the moon during the day sometimes.
I kinda understand intuitively, as the sun and moon obviously have different orbits, but I'd love to see actual visualisations of this, perhaps in 3D.
I find the more frustrating / noticeable the equation of time for a given point, where solar time and local time diverge, day to day it feels very counterintuitive, but when you see the chart it makes complete sense…
Yeah I have to say the north of Europe is bad folks. Winters are terrible. You just go to work/school in the dark and come home in the dark. There’s also less snow now than twenty years ago which means even less light.
In the summer it’s light outside for more hours than you are awake. Okay wow, so what? It’s a nice novelty. But it doesn’t make up for it. Not remotely.
Midsummer is a sad time because now I know that I will lose X minutes every day of light.
Using DST is just the extra bit of small cruelty on top of that.
Thanks for posting! It's a good read, quite Bartosz Ciechanowski esque
Oh, I learned from the graphs that today, March 20, seems to be peak at Additional minutes of daylight we get.
And that everywhere on the planet there is even split between day/night 12h/12h
Nice typography. Classy.
I knew days lengthened quicker around springtime, but your interactive visualization and clear breakdown of the math behind it makes this phenomenon tangible
(note - only semi-relevant)
While southern Florida is hardly at the equator, living here has really highlighted how northern-hemisphere and temperate-centric the online sphere - tech in particular - tends to be.
We don't get spring/summer/autumn/winter so much as rainy and dry season; heat pumps are irrelevant; natural disasters come in the form of hurricanes, and weather is either sunny or stormy; the days don't change in length much; and so on and so on.
It's a bunch of little things, but it's been surprising to me how often they come up in discussions, and just how rare (sub)tropical-specific problems and topics are in comparison. It makes me wonder what it's like to live somewhere even further removed from the natural world of the north.
All days are 24 hours except the one when we “make days longer” which is only 23 hours.
Then we “make days shorter “ resulting in a 25 hour day.
I always think of what time is sunrise where I am. Watching it get earlier/later every day by a couple minutes.
I love this - shout out from the Southern Hemisphere though - would love to see our version of it.
Unfortunate that the latitude is not allowed to be negative.
Use the derivative, Luke
It's always interesting how it affects how people think about the day. If you look outside and it's dark then it's easy to think that the day is basically over, even though the time might say otherwise.
I really appreciate the long winter nights and long summer days in Norway. Being able to wake up to the sun as early as 04 and enjoy it until after 23 is great, and the UV is only high during the middle of the day. Long summer days are awesome. Long winter nights are probably not as appreciated, but I enjoy those too.
July 5th through September in Seattle are a dream; incredibly long days, dry but not too hot. The sun never sets. I've been a lot of places on the planet and PNW summers are uniquely great.
However, during the winter it's dreary, it's dark by noon, and every year around October the raven comes tapping at my chamber door.
July of 2020 I left and went south to Central America. That first 'winter' I kept waiting for my annual S-A-D season to start. It didn't. The second 'winter' was worse because I reasoned the lst time could have been endorphins keeping me going. I was really expecting any day for the depression to start. And again, it didn't. December and January felt normal. That very consistent 7am it's bright sunny day and at 7pm it's dark night all year is helpful.
Quite literally, I never intended to return to Seattle, but with the recent economic turmoil it worked out that way. S-A-D season kicked off like usual, even with the 10,000 IUs of Vitamin D.
Without a job I moved to Eastern Washington, and within a month I was doing better, life got easier and the future was far less inscrutable. I have the energy to do the work, and after an angry or sad thought the follow-up thought is "that was a bit dramatic". That's a sign every year that things are on the up-swing.
Many people love and appreciate the Seattle climate just as it is; I don't know really anyone that reacts quite as severely as I do to it. Not being in Seattle weather means I get 6 more months of life every year. It kills me how much time I wasted trying to 'fix myself' in a place I just wasn't meant to be.
For the Americans, daylight savings time also changes the perception of how the days are getting longer.
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The writer had to attend a standup with a colleague in Norway to realize this and write an article. Funnily enough, as a Muslim I get reminded about this annually during Ramadan, which is right now.
First of Ramadan this year coincided with March 1, and it was a 12:45 hours of fasting from the first light of dawn to sunset at my location, also near Los Angeles. Today it's going to be 13:15 hours long, and by the time last of Ramadan rolls in around the end of March, it will be 13:37 hours.
Ramadan is observed following the lunar calendar, which is shorter than solar- based calendars by about 10 days. A winter Ramadan is short and easy in the northern hemisphere and we will have the shortest days in 2031. 2047 it's going to be middle of summer, so the hardest.
In case you ask, well what about places where sun does not set? When do you have your Suhoor (meal before dawn) and iftar (breakfast meal at sunset)? Opinions differ, but people usually follow the more realistic time of sunrise and sunset at a reference location. My brother in law was in Sweden few years back and he used the time of Mecca as reference.