One interesting consequence of the varying offset between mean noon and solar noon is that the when the days get longer / shorter around the shortest / longest day, the change in the morning isn't always the same as the change in the evening. In the southern UK, where I live, sunrise might continue to get later for a couple of weeks while sunset gets earlier. It took me ages to understand why.
Now, can you prove that the shortest night of the year is adjacent to the longest day? This is indeed the case for all latitudes, but it is wickedly difficult to prove.
Hint: start by setting up an imaginary planet with irregular orbit where this is not the case. Then find what property of the orbit you needed to force that does not happen with the earth.
I wrote something similar in Pluto.jl (Julia-native reactive notebook), but mine focuses more so on the apparent analemma from a given point on earth at a given time, rather than explaining the fundamentals of how an orbit gives shape to the analemma.
Interactivity requires either running Pluto locally in Julia (`using Pluto; Pluto.run()`) or hosted on Binder.
This is awesome work. I had tried to figure this out on my own but ran into a lot of trouble, and seeing everything together here finally made it click.
It's a combination of up-down (north-south) and east-west motions. The up-down component is simple, that comes from the axial tilt.
The east-west component arises from this: When the Earth is at perihelion (January), the time from solar noon to solar noon is slightly longer, because the Earth is moving faster and farther in its orbit, so it has to rotate a bit more until the same meridian points back at the sun again. At aphelion (July), that's reversed, solar noon to solar noon is slightly shorter.
Put another way: A solar day is more than 360° of rotation relative to distant stars. It's very close to 361°, because the angle between the stars and the sun changes as the Earth advances in its orbit, by about a degree per day. At perihelion in January, it's slightly more yet again, like 361.0002°, and in July, it's 360.9998°; that difference takes a few seconds more or less to rotate. We don't vary our clocks based on that variable solar day duration; instead we fix our daily time intervals and let the sun's apparent position vary east-west slightly relative to that.
The figure-8 shape arises from the relative phase of the two cycles. The maximum velocity for the north-south component occurs at the equinoxes, while the maximum velocity for the east-west component occurs at perihelion/aphelion. On Earth, these occur at different times. Interestingly, for Mars the converse is true; its equinoxes are near its perihelion/aphelion, and so Mars's analemma is almost not a figure-8 (it is, but one lobe is very small.)
It's a misconception that the figure-8 shape is due to the eccentricity of the Earth's orbit. Most of the variation in the sun's horizontal apparent motion is also due to Earth's axial tilt. Orbital eccentricity only contributes the slight asymmetry seen in the final analemma. See my reply to the sibling comment for more detail.
It would be purely up-and-down (due to the Earth's 23 degree orbital tilt) if the Earth was on a perfectly circular orbit. But because the orbit is slightly elliptical, and because its orbital speed thus varies throughout its orbit[0], the alignment of the Sun on the sky at "noon" drifts back and forth according to when the Earth's orbit "outpaces" its daily rotation or when the daily rotation outpaces the orbital motion.
This is incorrect. The analemma would still be a figure-8 even with a circular orbit, but it would be symmetric -- the orbital eccentricity only adds the asymmetry we see in Earth's analemma.
This is because the sun on an axial-tilted plane 'lags behind' then 'catches up' to the ideal 0-tilt sun over the course of the year. At the equinoxes, the sun's motion has a significant vertical component. Therefore, it's horizontal speed is slower than an untilted sun (both suns still travel through the sky at the same speed of 360/365 degrees per day), so it will lose ground and drift back. At the solstices, the sun moves horizontally, but at an higher latitude (equal to Earth's axial tilt) on the celestial sphere, covering more degrees of longitude for the same speed than the ideal sun moving along the equator, hence making up the lost ground. This variation in horizontal speed throughout the year creates the figure-8.
I'm having a difficult time understanding this. I trust that you're knowledgeable about the topic given your comment history, but --
You've argued that the Sun's rate of motion across the sky should have a constant magnitude (under the assumption of an Earth analog with a perfectly circular orbit but still 23.5 degree obliquity). It's not clear to me why this should be the case; on the celestial sphere, I would have an easier time accepting that the Sun should have a constant "horizontal" (i.e., in the sense of right ascension) motion due to the Earth's constant orbital speed, and that the Sun's "vertical" (i.e. declination) motion should be purely sinusoidal: greatest at the equinoxes, zero at the solstices. Have I misunderstood something here?
As an extreme case, imagine a tidally locked planet on a circular orbit of 1 AU with 23.5 degree obliquity. An observer sitting on the equator would see the sun moving on a line overhead, crossing the zenith back and forth, with no figure 8 -- right?
It is quite difficult to visualize and it took me a long time to "get it".
The path of the sun's annual motion relative to the stars is determined solely by my physical progress in orbit around the sun. A planet's axial tilt only changes the 'direction' I'm looking in and thus the reference point of my celestial coordinate system. The sun's path will always be a great circle on that celestial sphere (and always the same relative to the fixed background stars) regardless of which reference frame I choose. I think this is enough to surmise that the sun's angular speed on the celestial sphere is constant regardless of axial tilt (assuming a perfectly circular orbit).
Taken to an extreme, imagine a planet with 90° tilt -- the sun would move vertically and pass directly over the pole, making a constant 'horizontal' motion literally impossible.
I'm not sure what your tidally locked example is meant to demonstrate, since that's literally what the analemma is -- the path the sun would make in the sky once you subtract a planet's local axial rotation, i.e., tidally locking it to the orbital parent.
Thank you for the further explanation - this has convinced me that it's plausible, though my mind is slow and I still need to think about it to be fully convinced.
In Neal Stephenson's "Anathem" an analemma observed on an alien spacecraft leads the planet's inhabitants to speculate that the spacecraft is from ****** (not going to spoil it).
