Elevation difference of analemmas' taken at different hours and the reason for the tendency of analemmas' being slanted with deviation from the noon

Question

According to Wikipedia, in astronomy, an analemma (/ˌænəˈlɛmə/; from Ancient Greek ἀνάλημμα (analēmma) 'support') is a diagram showing the position of the Sun in the sky as seen from a fixed location on Earth at the same mean solar time, as that position varies over the course of a year. The diagram will resemble figure eight as shown in the following picture which is an afternoon analemma photo taken in 1998–99 in Murray Hill, New Jersey, USA, by Jack Fishburn.

The discrepancy between the true and “mean” Sun position during the course of a year resulting in this figure-8 pattern, or analemma can be photographed on a single photographic film by keeping a camera (a sun filter can be used to avoid excessive light) at a fixed location and orientation and taking multiple exposures throughout the year, always at the same time of day (disregarding daylight saving time).

The average height(elevation) and the maximum height(elevation)[which may not be the correct technical word] (the top point of the analemma) of the sun at 8 AM should be lower than say, that of an analemma photographed at noon (which is around the solar noon but not usually the same precisely). This is because the sun continues to rise in the sky after 8 AM to reach its highest point around solar noon. Therefore, when you compare the two analemmas, the one photographed at 8 AM each day throughout the year should have its top point lower than the one photographed at noon each day throughout the year. (For that matter this usually means the observed mean elevation of analemma at noon shall be higher than the analemma in the morning or afternoon)

But such an effect can not be discerned or represented (apparently) in the following photographs of analemmas which are taken at various locations in Greece over different hours. https://www.perseus.gr/Astro-Solar-Analemma.htm

(Of course, they are slanted at different angles and one can argue that this can generally imply that the top point of the slanted may be lower than that of the vertical one and that photographs are not of the same scale.)

I have two questions:

What is the reason for this discrepancy in scale? (I mean, is the vertical one [or the other analemma around noon] of the same apparent elevation as others which are photographed at earlier hours or is it due to photographs not being in the same scale or some other reason?)
Analemmas photographed during earlier or later hours tend to slant. I think this implies that the clock effect on the observed position of the sun (due to solar time being somewhat different than that of time represented by our clocks) is stronger at earlier or later hours than at say, noon. What is the reason for this?

Not a direct answer to your question, but this site explains, and lets you alter, the properties of an analemma: mtirado.com/blog/demystifying-the-analemma — Greg Miller, Commented Jan 9 at 17:30

notovny · Accepted Answer · 2024-01-09 12:13:17Z

4

Partial answer:

With regards to the scale of the analemmas in the pictures in Greece, it's basically that the analemmas are the same angular size regardless of where and when you photograph them from Earth, but the photographer both wanted particular landmarks in the resulting photograph, and needed a place near the landmarks in question where they could place a camera for a year, and those constraints defined the field of view of each photograph.

As far as the angle of the observed Analemma goes, it's a function of the axial tilt of the Earth and the latitude observer. If the Earth's orbit was a perfect circle, the analemma would be a segment of a great circle perpendicular to the celestial equator, and for any position on the surface of the Earth other than at the poles, that line appears "slanted" away from local Noon because that arc would appear "slanted" from that location.

edited Jan 9 at 12:13

answered Jan 9 at 11:09

notovny

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$\begingroup$ Those animations are great. :) But I don't understand why direction marks are changing in the background. If the background is the sky, why do E and W interchange? And E and W directions should be horizontally aligned and N and S should be vertically aligned. Shouldn't they? $\endgroup$
– Jacob Miller
Commented Jan 9 at 12:37
1

$\begingroup$ The green half is the ground, the blue half is the sky. The animations are based on where the analemma wound be for an observer at the specified location, if taken every 24 hours from January 1st at the time in the title, and centered on the position of the Sun on March 20th at the time in the title of the graph. The reason it flips for the Ecuador image is because it's only showing a portion of the sky as it looks at the position of the analemma for that daily time, and tries to keep the ground on the bottom of the image. $\endgroup$
– notovny
Commented Jan 9 at 13:08
2

$\begingroup$ The compass directions change because the animation tracks the Sun for a full 24 hours, and you watch the Sun move from noon (almost due S) to setting (W) to lowest point below the horizon (N) to rising (E). The compass directions are shown wherever the line of constant azimuth passes through the edge of the image. In other words, if you are looking at something 45 or 80 degrees high, you are still facing some direction (S, NW, etc). The lines that are mostly vertical and converging at the zenith are the azimuth lines. $\endgroup$
– JohnHoltz
Commented Jan 9 at 17:43
1

$\begingroup$ At the North Pole, all directions are technically South. You cannot go or look East when at the North Pole. Skyfield somehow picks one direction to be South which then allows all other compass directions to be labeled. $\endgroup$
– JohnHoltz
Commented Jan 9 at 17:45
1

$\begingroup$ @JacobMiller It's a major factor, yes. There's also the fact that the analemma isn't lengthwise-symmetrical because the elliptical orbit of the Earth, but the main reason for the tilt in most analemma images is a combination of the latitude, and of the local time chosen to take images of the sun. $\endgroup$
– notovny
Commented Jan 10 at 22:57

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Jacob Miller · Accepted Answer · 2024-03-01 00:53:54Z

This answer was written thanks to the information shared by PM 2Ring, notovny and JohnHoltz.

So in essence those photographs don't represent visual altitude differences of the analemmas at different hours for the location where they were taken.

In addition to what notovny's answer already explained, the photographer's focus on representing the same angular magnitude of the analemmas' on same-sized photographs and the constraints of involving the projection of a 3-D spherical shape of the sky to a 2-D plane of a photograph may have defined the resulting view.

No, the slant of analemma is not due to the clock effect being stronger at certain hours or times on the observed position of the sun, but rather due to

the deviation of the North-South axis of the analemma from the observer's (arbitrary) reference line at an acute angle
a combination of Earth's axial tilt of 23.5$^{\circ}$ relative to the ecliptic (Earth's orbit around Sol, the name was coined because eclipses happen along this plane) and ecliptic being slightly elliptical

Or in simple words, as user "notovny" stated, analemma appears slanted because of the perspective of the observer plus the asymmetry around the North-South axis. (which factor affects how much will depend on at which mean solar time of the day for the observer's longitude the analemma is observed or recorded, and the latitude of the observer's location)

The nominal perception of the casual layman about the Sun's daily movement is that it rises eastward and sets westward no matter where the observer is located. More astute and knowledgeable of these laymen may also comprehend that this daily motion would not be observable at extremities of the North and South poles where the Sun would appear in an annual cycle of movement. However, if they were to actually observe the Sun’s position at the same time throughout the year, they would not see it move along a line, instead, would see a figure-of-eight curve or at least a portion of the curve, in the sky, wherever the observer is at Earth irrespective of the daily time at which the Sun's position is recorded.

A solar analemma can be shown on a graph where the declination of the sun is plotted against the difference between (apparent) true solar time and mean solar time, at the same mean solar time each day, for enough, regulary close days (in order to show correct shape) throughout the year. The resulting figure is similar to a slightly distorted number eight shape of which the upper loop is overly small.

In this plot, the axis of declination also represents the axis of mean solar time since the aforementioned difference becomes zero along it. And as mean solar time by definition, the average of apparent solar time, magnitude of total difference right side of the plot shall equal the left side of the plot although opposite signed. Hence axis of declination is also a lengthwise mean axis of annalemma. This is the axis which is of special interest to the context of this answer.

Furthermore, as the declination (of the Sun) is the apparent North-South movement of the Sun, this axis of interest is also aligned with North-South hence by definition with the axis of rotation (of Earth). As the equatorial coordinate system is defined as a projection of spherical coordinates of Earth onto the celestial sphere, the North-South axis will be same for the both.

This lengthwise mean axis of the analemma may hereinafter referred to as the North-South axis or the axis of declination interchangeably for brevity or the sake of clarification.

Since 24 hours is the mean day length, the mean solar position straight over the observer's local meridian (known as the culmination of the sun or commonly as noon) corresponds to 1200 hours (mean solar time) by convention. Furthermore, as Meridian itself is (as well as any celestial line of longitude, which inherently are also Meridians) North-South aligned, it also contains the axis of declination of the sun. Consequently, Meridian can also be considered to contain the North-South axis of the analemma.

The difference between (apparent) true solar time and mean solar time is also called the "Equation of Time".

^^^^^^ Approximate Equation of Time (in minutes) - image credit: PM 2Ring ^^^^^^

The apparent solar time directly tracks the diurnal motion of the Sun (East-West movement of the subsolar point). In contrast, mean solar time tracks a theoretical mean Sun with uniform motion along the celestial equator. The apparent solar time can be determined by measuring the current position, or hour angle, of the Sun as indicated (with limited accuracy) by a sundial.

Because human activities needed uniformly conformed standard of time, mean solar time was adapted as Local Mean Time (LMT) from the early to late 19th century in which, longitude-specific solar mean time was used (each location -town, city etc.- kept its own meridian specific time) until the Standard Time which incorporates regional times zones, where same time is used throughout the same time zone despite earlier used LMT being different across the longitudinal span covered by it, was adopted as the civil time. These time zones were calibrated using + or - offset from GMT (Greenwich Mean Time) which was the universal Time (UT). Nowadays UT is calculated by reference to distance celestial objects. A high-precision atomic coordinate time standard, International Atomic Time (TAI) is the weighted average of the time kept by over 450 atomic clocks in over 80 national laboratories worldwide. UT coordinated with, TAI adjusted by leap seconds to compensate for variations in the rotational velocity of the Earth, forms UTC which is the modern international standard from which offsets to other time zones are calculated. GMT corresponds to UTC+00:00.

Upon the Standard Time based on UTC, another adjustment named Daylight Saving Time (DST) was adopted for efficient daylight usage as the name suggests, in some regions or countries for part of the year where significant change in daylight hours occurs as per season. To avoid tangling in this complexity, solar mean time signifies its genuine sense, the Local Mean Time or LMT in this answer unless otherwise noted.

This difference between apparent solar time and mean solar time is due to a combination of Earth's axial tilt of 23.5$^{\circ}$ relative to the ecliptic and ecliptic being slightly elliptical. (There are other factors contributing to this difference but they are practically insignificant to our short-term civil experience of time and solar movement, and hence are safely disregarded in the context of this answer.)

If taken alone, the clock effect only due to the tilt of the earth's rotational axis from the ecliptic (earth's orbit around the sun) would have resulted in a figure eight shape with both lengthwise and crosswise symmetrical lobes. This means that "figure of eight" would have had the same sized and shaped upper and lower lobes with the left and right sides being symmetrical around the lengthwise mean axis that represents the declination of the sun.

However, the additional clock effect due to the earth's orbit being slightly elliptical caused one lobe to be significantly smaller than the other and the shape of figure-eight being slightly skewed to the side (slightly slanting or tipping to the side) of the lengthwise mean axis that represents declination or North-south movement of the Sun. This also causes a slight disfiguration of the figure-eight shape. (To see a visual comparison see pages 25 and 27 in chapter 4 of The Analemma For Latitudinally Challenged People by Helmer ASLAKSEN, Shin Yeow TEO)

Analemma plotted as seen at noon GMT from the Royal Observatory, Greenwich (latitude 51.48° north, longitude 0.0015° west). Please note that the analemma is plotted with its width highly exaggerated, which permits noticing that it is very slightly asymmetrical.

However, it is not the "clock effect" that causes the "slant" (used in the sense of, deviation at an acute angle from alignment with a reference line, which in this scenario can be either the observer's local horizon or at the equator, the celestial equator) of the analemma's North-South axis (which represent mean solar time as previously mentioned), which can be observed at early or late hours from locations other than the geographical poles or the equator.

This apparent slant of the North-South axis of the analemma means that it is oblique to the observer's arbitrary reference line which in this case usually is the observer's horizon, also called the local horizon or celestial horizon unless the observer is at the equator, where celestial equator is a more suitable reference line.

(Note that the local horizon is different from the celestial equator unless the observer is centred on either the North or South pole. The celestial equator is orthogonal to the North-South axis of the analemma.)

This slant occurs because the meridians (within the observer's local horizon) that are away from the local meridian appear oblique from the perspective of the observer. This acute angle (when referenced from where the meridian contact horizon) varies from meridian to meridian, between the angle corresponding to the latitude of the observer and the angle of the local meridian with the local horizon of the observer. As the local meridian itself is orthogonal to the local horizon at the reference point, this latter angle is 90$^{\circ}$. These consequences are due to the inherent geometrical nature of the spherical coordinate system.

As explained earlier these meridians contain analemmas' North-South axis. Hence when, the observer is located between 0$^{\circ}$ and 90$^{\circ}$N or 90$^{\circ}$S latitudes (which corresponds to the locations between the equator and, poles), analemma's North-South axis becomes oblique from the observer's local horizon. This is the apparent slant of the axis of declination that we notice when analemma is recorded in the morning or evening hours between the poles and the equator. However, at 1200 hours LMT, the analemma's North-South axis is along the observer's local meridian which is the semi-circle that contains the observer's zenith(the point in the sky that is directly above the observer, or the opposite of the nadir) and North and South points of their local horizon. As the observer's local meridian is orthogonal to the observer's local horizon (at the reference point), analemma's North-South axis itself becomes perpendicular to the (tangent of) local horizon. So the analemma's North-South axis is observed un-slanted.

But the earlier mentioned inherent slight skewing of the analemma from its North-South axis causes its figure eight shape to appear slightly slanted whenever its North-South axis appears unslanted. As a result, the figure-eight shape looks truly unslanted only when its axis is slightly slanted which can only happen for analemma's observed slightly before or after the 1200 hours LMT.

When at the equator, the observer's latitude is 0$^{\circ}$. Hence, meridians within the local horizon are oblique at a range of 0$^{\circ}$ to 90$^{\circ}$ to the reference point. However, the usual reference line, the horizon is not very sensible here (especially when it is at the zenith where the horizon is farthest) as the analemmas will be in the center of the sky for a significant time. There is a more intuitive reference line that the observer is likely to use at the equator. It is the celestial equator which bisects the local horizon and observer's sky dome when observed from the equator. Now by definition celestial equator is perpendicular to the axis of rotation hence to the North-South axis of the analemma. Hence for the equatorial viewer, the analemma rises from the East proper, travels with its crosswise mean axis centered right on the meridian and sets West exactly, all the while its North-South axis being orthogonal to the celestial equator which passes through the zenith. Again, the inherent slight deviation of the figure-eight shape of analemma from its North-South axis causes it to be always slightly slanted even for the equatorial observer.

On the other hand, at the poles, the observer's latitude is 90$^{\circ}$ which means all meridians and as a result the North-South axis of analemma at any time shall be orthogonal to the horizon. We can also deduce this in another way. As the exact poles are antipodal points where the circumference of the Earth contacts its rotational axis, all meridians crisscross, resulting in the local meridian becoming an identical multiple converging at the zenith for an observer located at either of the poles. As mentioned earlier, as the observer's local meridian and observer's (local) horizon are orthogonal, and analemmas' North-South axis is contained in meridians, whichever the time is, analemma's North-South axis is observed unslanted (orthogonal to the local horizon, aligned with the local meridians) during the 24 hours of the day though only a portion of the analemma is visible. However, the figure-eight shape will always appear slightly slanted for observers located on poles due to its natural deviation from its North-South axis. So for the polar observer analemma will revolve around 360$^{\circ}$ for different times of the day, slightly slanted from the meridians but neither rising nor setting.

Please note that analemma revolving, rising and setting refers to different analemmas taken at different times of the day, not a single analemma moving or revolving around. "Solar Analemma" is a record or composite image of the sun's position in the sky throughout the year taken from the same observing location at the same exact mean solar time specific to the exact longitude of the location.

Summary

So basically, the slanting of analemma when observed at different times of the day is a result that is dictated by the relevant rotational and orbital properties of the Earth as well as laws of spherical geometry in combination with the observer's location and time.

And also the light (as light tends to travel in straight lines, otherwise the shape of analemma could have easily been different). Then again all the physical properties of the universe and limits of human perception would have to be listed (please disregard strikethrough sentences as they are over the line of the limited context of this post and the answer. They were included to give an appraisal for the casual reader about the contextual limit this explanation addresses.)

Furthermore,

Analemma or figure-eight shape will only appear unslanted (relative to the reference lines considered) when observed from between the equator and the poles of the Earth, and never at noon in the Local Mean Time (LMT).
In contrast, however, the North-South axis of analemma (which represents mean solar time and declination of the Sun) will always be unslanted for observers located at the equator or the poles. For observer's between latitudes 0${^\circ}$-90${^\circ}$N or 0${^\circ}$-90${^\circ}$S this axis is unslanted only at 12:00 am local mean time. Note that this axis is never visible but deduced as the declination or sun's apparent movement parallel to the Earth's axis of rotation.
In addition, as standard time zones cover a vast span of longitude, standard clock time may not represent the correct mean solar time for the exact location of the observer. So, this explanation utilises Local Mean Time (LMT) which represents accurate mean solar time for the observer's exact longitude.

(It is important to note that analemma on other planets can appear in a different shape than on Earth depending on the determining factors.)

For more info and an in-depth analysis of analemma please refer to the research paper "The Analemma For Latitudinally Challenged People" by Helmer ASLAKSEN, Shin Yeow TEO which will provide a comprehensive understanding of analemma for the latitudinally challenged casual layman as the title says. :-)🙂 Also https://www.analemma.com/ explains analemma with visually appealing easy-to-understand animations for the novice.

For an interactive simulation of analemma please see the following answer by user PM 2Ring: https://astronomy.stackexchange.com/a/55736/54451 Note that if you want multiple analemmas for different hours on the same image change the time step to hours (h) value. For example change "7d" in default configuration to "23h" or "25h" to see analemmas for all the hours of the day in the same image. Changing to other values may give a different number of analemmas. So adjust at your will and see the results. Transparency can be changed by modifying Python code. (add opacity=0.5 to the function call that creates the globe: G = pp(xyz(u, v), ... )

A very useful analemma calculator can be found at the mtirado.com. It has custom parameter input fields, 2D and 3D views and presets for Earth and Mars.