OT - Length of day question


Ok, this has nothing to do with calculators, even less with HP, but it has to do with math and physics and this forum has a bunch of very knowledgeable people, so I thought I might ask...
The shortest day of the year (meaning, the day with least sun illumination) is the day of the winter solstice, which is around December 21st. And I always thought that approaching that date, the sunrise would get always later and later, and the sunset always sooner. But I found out that the sunrise has its "latest" peak after the end of December, while the sunset has its "soonest" around December 13th.
I always thought they would be "symmetrical". I can't understand why the sunset stops getting earlier while the sunrise is still getting later. Does this question make sense at all? :) Anyone here knows the answer?



The rotational axis of the earth is not perpindicular to its orbital plane.

NOAA has (had?) a sunrise/sunset table calculator which one can easily plot to see the assymmetry.



Look up "Analemma" to learn more about this.

Essentially, the location of Local Apparent Noon -- the zenith of the sun--is not in the same place all year--and it is not symmetrical with the seasons.

Note that in addition to the tilting of our axis relative to the orbital plane, we also have a non-circular orbit. The northern hemisphere winter is actually when we are *closer* to the sun!

Obviously not only the tilt angle of our axis, but also where it points relative to the orbital apogee and perigee affect how our days get aligned.

For fun, see this:


Edited: 13 Dec 2011, 2:21 p.m.


As Bill notes, the Earth's orbit is not circular.

As Kepler deduced about 400 years ago, all orbits are ellipses, with the Sun at one of the foci - hence the Earth's distance from the Sun varies throughout the year. He also realized that the speed of a planet in its orbit is not constant, but when the planet is closer to the sun, it moves faster and vice-versa.

However, the spin rate of the Earth - i.e. the length of the day - is basically constant. (I say basically, because at the millisecond level, the length of the day does vary.) This is the length of the day that you measure with respect to the stars or something far, far away from the Earth.

A "day" as most of us think of it, however, is tied to the Sun. i.e. It's a "day" from one noon to the next, say.

But, since where the Sun is in the sky depends on both the rotational (daily spin) position AND orbital location of the Earth, the time of local solar noon changes systematically throughout the year - the curvature of the Earth's orbit has to be taken into account and added to (or subtracted from, if we are getting behind) the Earth's rotational position.

This effect shifts the daylight part of the day ahead or behind. This results in the analemma effect (the figure 8 often drawn on a globe) which notes how far behind or ahead the sun is. The amount of behindedness or aheadedness is with respect to a fictitious Sun that would be observed if the Earth was in a circular orbit.

To a small extent, the shift is fast enough that part of it occurs during the day, shifting sunrise and sunset by slightly different amounts. As also noted, the tilt of the Earth's axis, which affects where the Sun appears to rise and set, also comes into play.

See here for some more details.

also .

Edited: 13 Dec 2011, 3:15 p.m.


Hi Bill,

Is there any RPN program (I suppose someone already did it) calculating the equation of time? Since I cannot afford a watch with such "complication", at least I could find consolation programming this on my 34s :-)



Oh yes - the HP-65 Navigation Pack had such a program which had to run in 99 steps and less than 10 registers so almost any more recent machine should be up to it. The algorithm was a sum-of-harmonics approximation as I recall. It's been a long time since I read my copy of that 1975 era book...



I don't have an RPN program, but it should be pretty straightforward to go from the NOAA equations to an RPN program. The equations will get you from local Lat/Lon and time (with UTC offset) to solar azimuth and elevation and sunrise/sunset times.

For those of you using industrial Programmable Controllers, I have a version in IEC-61131-3 structured text language available as well.

There are about a dozen program constants, and lots of sin/cos/tan and asin/acos/atan. Probably too many steps for an HP-25, but I think an 11C could handle it.



Hi Miguel

Is there any RPN program (I suppose someone already did it) calculating the equation of time?

cf. article 1014: Sunrise and Sunset

Kind regards


PS: If interested in the math I highly recommend the paper by Christian Blatter.


Hi Miguel,

See my response below.

BTW, for those that have the museum DVD there is a 41 program there as well.



Edited: 13 Dec 2011, 9:25 p.m.


Thank you all for your reponses.

Now I have a nice and interesting reading ahead.




I always thought they would be "symmetrical". I can't understand why the sunset stops getting earlier while the sunrise is still getting later.

Good remark!

A lot of people actually have understood that sunset and sunrise changes would be symmetrical. The main reason is that a lot of us have been teach like that, since 'length of day' have only been explain through season inclination of the Earth’s axe of rotation.
If we only take into account this only explanation, then the sunset cannot stop getting earlier while the sunrise is still getting later. Something important is missing.

The important fact is missing!

Edited: 13 Dec 2011, 3:50 p.m. after one or more responses were posted



The shift has nothing to do per se with the sun being a disk nor with our definitions of sunrise and sunset. I am not saying that these do not have an effect on what we consider a "day" but the shift exists regardless of the size of the sun's disk...

By the way, when you see the sun just kissing the horizon, it is actually pretty much all the way below the horizon--because the rays are bent, we see the sun over the horizon. In other words, there is essentially a superior mirage at every sunrise and sunset...but this can vary dramatically with weather conditions. Timing the sunset can be quite off from the theoretical point.


You right, I miss samething in my explanation.

And only realise now that analemma have effect on sunrise and sunset.
Up to now, I only consider analemma to have impact on position of the Sun in the sky and only care of it for navigation application.

I like this forum. Every day something to discover and misskownlege to correct.

Thank you very much.


Hi C:

"I like this forum."

Me too! :-)



Hi C:

"I like this forum."

Me too! :-)

Me three! :)


A sunrise/sunset table for a year can be generated here.


I have a pretty accurate sunrise/sunset program for the HP-41 (and versions for several other calculators such as the 33S, 35S, and 39gs among others). If anyone is interested please contact me using forum email.

Relatively slow on the 41C/V/X but runs great on the 41CL.



Edited: 13 Dec 2011, 3:54 p.m.


Here in the southern hemisphere 32 degrees South the pattern of the above is reversed. The earliest sunrise is December 4 and the latest snset is January 5 in the southern summer. in winter the earliest sunset is on June 5 and the latest rise is on July 6.


Thank you everyone for all the responses! It will take some time to read (and understand) all the material, but this is exactly what I was looking for. I knew I would get informative responses here... How I love this place! :)



I'll try to add some more information. Since the Earth orbit is an ellipse, according to the second Kepler law, the Earth angular velocity is greater when the Earth is closer to the Sun in its orbit (perihelion), and lesser when it's around the aphelion, because its surface velocity has to keep constant along the whole orbit.

The duration of the day is the time interval between two consecutive culmination of the Sun. Since the distance between the Earth and the Sun is not infinite, that interval consists of the time lenght of a whole Earth rotation plus an additional angle that is equivalent to the apparent motion of the Sun along the ecliptic. The angular velocity of the Earth along its orbit is greatest at the end of December, then the time lenght between two consecutive culminations is extended.

Hope it helps



I have added my programs for the HP41C, HP42S, HP33S, and HP35S to the articles section. For those interested in the HP39gs version please continue to contact me via forum mail.



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