[HN Gopher] Sinusoidal Sunlight
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Sinusoidal Sunlight
Author : surprisetalk
Score : 85 points
Date : 2024-10-23 21:31 UTC (1 days ago)
(HTM) web link (leancrew.com)
(TXT) w3m dump (leancrew.com)
| froggerexpert wrote:
| The sunlight plot is interesting.
|
| Since Dec wraps around to Jan, you can fold the left and right to
| make a tube.
|
| Since 23:59 wraps to 00:00 you can fold the top and bottom of the
| tube, making a torus (a donut).
|
| For a fixed lat/long, each point on the torus corresponds to the
| sunlight observed at a particular time throughout the year. Why
| bother with a torus? The shape itself embeds the continuity of
| time across days/years that is otherwise left implicit in the
| typical 2D plot.
|
| I've wanted to plot this in 3D or have it printed on a ring, but
| never got round to it.
|
| Any one seen anyone do this?
| jagged-chisel wrote:
| So a toroidal illustration of our trip around the sun with the
| "amount of sunlight" graph along it ... someplace.
|
| Sounds neat!
| froggerexpert wrote:
| Yes, exactly!
| jpm_sd wrote:
| Related:
|
| https://en.wikipedia.org/wiki/Analemma
| Centigonal wrote:
| You got me curious, so I gave it a try with the graphic in the
| article!
|
| https://www.loom.com/share/5665143f2d274bd0bf65ef378fad39a3
|
| There's two toruses in the clip, one with the daylight on the
| inside, one with the daylight on the outside.
|
| One thought I had while making this is that you could visualize
| multiple years, or even someone's whole life, as a string
| winding a long spiraling path down the length of a helix.
| froggerexpert wrote:
| Very cool. What program is that?
|
| It'd also be nice if the colour was not just day/night, but
| the actual predicted daylight at the time of day, which would
| result in a continuously changing colour.
|
| I guess at that point, the sine approximation from OP would
| no longer apply, and
| https://en.m.wikipedia.org/wiki/Sunrise_equation would have
| to be used.
| enriquto wrote:
| 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 wrote:
| 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 wrote:
| 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
| jvanderbot wrote:
| 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 wrote:
| That's true. It offsets both sunrise and sunset the same
| amount.
| jvanderbot wrote:
| 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.
| GistNoesis wrote:
| I find the wikipedia article
| https://en.wikipedia.org/wiki/Sunrise_equation more complete as I
| was wondering about latitude and the fact that north poles
| sometimes don't even see the light for days, a sinusoid wouldn't
| fit.
|
| Now wondering how accurate a location we can get from the
| observation of sunrise and sunset from the formula (in the case I
| got stranded on a desert island :) ).
| madcaptenor wrote:
| If you look at sunrise times at places just south of the Arctic
| Circle it's pretty obvious that day length is not exactly a
| sinusoid. See for example Reykjavik:
| https://www.timeanddate.com/sun/iceland/reykjavik
|
| Ignoring refraction, you have cos(omega_O) = - tan(phi) *
| tan(delta), where:
|
| - omega_0 is the hour angle at sunrise/sunset (basically the
| time)
|
| - phi is the observer's longitude
|
| - delta is the sun's declination, which varies over the year.
|
| delta is not exactly sinusoidal but that doesn't seem to be the
| major problem.
|
| The hour angle at sunrise is
|
| omega_O = arccos(-tan(phi) * tan(delta))
|
| and if delta varies sinusoidally then I think we can wave our
| hands and say "small angle approximation" to get an approximate
| sinusoid out the other end, but if tan(phi) gets large enough
| the approximation breaks down.
| knappa wrote:
| Instead of trying to fit a sine wave this way, one can also take
| the Fourier transform and read off the largest value and its
| location.
| searke wrote:
| You might want to use Fourier analysis instead. I did this
| project but with temperature :
|
| https://searke.github.io/2017/01/08/TemperatureGraph.html
|
| Also, coincidentally enough, for Chicago using Wolfram Language.
| Great minds think alike I guess.
| phkahler wrote:
| >> You might want to use Fourier analysis instead.
|
| Yep. There error plot clearly shows a 3rd harmonic dominating.
| IIRC the length of day is constant at the equator and the
| light/time function is a distorted sinewave as you go north or
| south from there. I am curious about the slope within a day. It
| seem to "get dark quickly" these day vs July.
| url00 wrote:
| What a great title! Sending warmth to you this winter from
| Wisconsin.
| PaulHoule wrote:
| My first take is that you can represent any function like this
| with a sum of sinusoids as it is a periodic function and
|
| https://en.wikipedia.org/wiki/Fourier_analysis
|
| applies. Note the error looks a lot like a higher frequency
| sinusoid and if you added in that high frequency sinusoid the
| error would look like a much higher frequency sinusoid.
| Traditionally people who did celestial mechanics and positional
| astronomy would expand everything in terms of sinusoids until
| they got good enough accuracy.
| jmward01 wrote:
| > You can decide for yourself whether an error like this--7
| minutes out of 9 hours--makes the real data "close" to a sine
| wave.
|
| I think the most interesting part of this article is the bit at
| the end. What really is 'close'? We have so many 'rules of thumb'
| here but a real definition to target is elusive. Do you go off of
| pure utility? 'using this definition achieves my requirement to
| be corret xxx% of the time and now I make money using it' Or do
| you go off of something more like information content: 'when
| plotting the error it conforms to a normal curve...' Anyone got a
| good rule for 'close'?
| abnry wrote:
| I don't think there will ever be an answer to this. There's a
| similar problem with choosing a threshold for a decision. The
| common thing to do is to make a RoC curve to compare the trade-
| off between true positive rate and false positive rate.
|
| If you have a system you can reason about completely, then
| sometimes you have a number that gives the absolute answer. Say
| you get error below floating point resolution.
|
| But I guess it's otherwise what is perceptible or meaningful,
| either in quantity or percentage. A penny is not a life
| changing amount of money and something happening 0.01% of the
| time is rare enough to be tolerable.
| amenghra wrote:
| "The error varies in a sort of sinusoidal fashion" and if you.
| Approximate the error with a sin, you'll get another sort of
| sinusoidal delta, and so on. The process is essentially a Fourier
| transform.
| mturmon wrote:
| OP did allow the base frequency to vary when he did the fit. So
| if you found the first residual and then used the same varying-
| frequency fit, you might not get an exact harmonic of the base
| frequency. That would not be the Fourier transform!
|
| But suppose you fixed the base frequency (which wouldn't be a
| bad idea). You still seem to have a nonlinear fit, because the
| phase (the "f" in the model equation in OP) is buried inside
| _sin()_. Why are we needing a nonlinear function-fitting
| process when the Fourier transform is linear?
|
| Of course, you can bring the phase outside by adding in a
| _cos()_ term with its own amplitude. Now instead of the phase
| "f" you have an interplay between the amplitude of the _sin_
| and _cos_ terms, and those amplitudes are linear in the data.
|
| The resulting fit (or recursive sequence of fits) would indeed
| be the Fourier transform.
|
| The key property is the orthogonality of the various harmonics.
| That's what allows the sequence of single-frequency fits to not
| step on each other.
| dakiol wrote:
| A bit off-topic but I couldn't overlook that Chicago reminds me
| of Munich (in terms of daylight).
|
| I have lived in Spain (Santiago de Compostela) and I absolutely
| loved that in the summer time the sun sets around 10pm. Even in
| winter time the sun sets around 6:30pm. I have lived in Munich,
| and it was depressing as hell in winter because the sun sets at
| around 4pm.
|
| I also hated that in summer in Munich, the sun rised around 5am.
| I'm not a morning person, I never cared for how much daylight I
| was getting before 9am (which is more or less the time I wake up)
| Gare wrote:
| Major reason sun sets so late in Spain is because they're in
| the "wrong" timezone.
|
| Daylight in Santiago is only 40 minutes longer during winter
| solstice. And during summer solstice Munich has longer
| daylight!
| ncruces wrote:
| Particularly Galicia (Santiago), which should really use the
| Portuguese timezone.
|
| Or not: you cross the border and don't "fix" your watch,
| because mealtimes, etc, are all shifted an hour in the
| opposite direction.
| themaninthedark wrote:
| And this is why we have daylight savings time...
| crackalamoo wrote:
| I wonder if this discrepancy can be explained by the equation of
| time? https://en.wikipedia.org/wiki/Equation_of_time
| antognini wrote:
| Last year I made an interactive plot so that you can see how much
| the length of a day changes over the course of the year as a
| function of latitude. It goes through the math of how you can
| make the calculation.
|
| You can check it out here: https://joe-
| antognini.github.io/astronomy/daylight
| hprotagonist wrote:
| The analemma graph shape happens because you're adding sinusoids:
| https://en.wikipedia.org/wiki/Equation_of_time
|
| See also, https://www.gaisma.com/en/ for some really good
| interactive sunlight graphs.
| SideQuark wrote:
| > how close to a sine wave is the Daylight line?
|
| It's as close as the Earth's orbit is to a perfect circle and
| that the shapes of the earth and sun are to perfect spheres and
| as the axis of the earth has no wobble and no precession and the
| sun's rays are truly parallel and and there is no drag in space
| sapping energy from the dynamics and general relativity effects
| are sent to zero and so on.
|
| Once everything is perfectly circular and simple and Euclidean,
| then it does become a perfect sinewave. Then the sinewave has
| period = 1 year, the amplitude is symmetric by latitude, and it's
| pretty easy to derive.
|
| All the above (and more) all intorduce measurable errors from the
| perfect sinewave.
| rachofsunshine wrote:
| Or for those who prefer their deviations from ideal behavior in
| pretty curve form, the analemma [1] captures the way in which
| the Sun's fails to return to the same point in the sky each
| day:
|
| - The north-south component comes from the Earth's axial tilt
|
| - The asymmetry from north to south comes from the elliptical
| nature of Earth's orbit, which is closer/faster in northern
| winter + further/slower in northern summer.
|
| - The east-west movement comes from the equation of time, which
| is mostly a mix of the Earth's axial tilt and the eccentricity
| of its orbit.
|
| [1] https://en.wikipedia.org/wiki/Analemma
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