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I'm finding several sources on the internet which say that the solar analemma at the equator is a horizontal line, but this makes no sense to me, especially the analemma at noon, for which the sun should surely travel vertically from the tropic of cancer to the tropic of capricorn and back (and probably with some some lateral displacement as well.)

I'm expecting that the dawn and dusk analemmas are far closer to horizontal, but there should still be some shape to them -- a simple line does not seem likely. The best answer from my point of view would show/describe 15 analemmas, representing the sun's positions through the year as seen from both tropics and from the equator, and pinpointing the locations at 7am, 9:30am, noon, 2:30 pm, and 5 pm.

An extension of this question (Please note: I will use the singular form “analemma” because it seems easier to understand, though “analemmas” would be more correct: it is not a single object but a set of human constructs plotted over time and space.)

Thanks all for your comments on the original question. Most seem to favor the horizontal orientation, either a simple east-west line or a horizontally oriented infinity sign, an eight lying on its side. My sense is that these images may not be grounded in empirical observations of how the analemma (or its shadow on the ground) actually appears in the equatorial zone, but instead derive from thinking about why the analemma appears as a bottom-heavy figure eight in the northern hemisphere, compared to a top-heavy eight in the southern hemisphere.

If anyone is aware of scientific observations that confirm the idea of an east-west orientation at the equator, please let me/us know. Otherwise, I'd like to share an alternative answer which I find more believable. Of course, my answer too is merely based on pondering why the analemma is reversed in the southern skies – I have no experimental observations of my own.

I don’t see how these “horizontalization” solutions could work. They beg the question: At what point does the analemma’s northern hemisphere verticality flip to being horizontal, and at what point does this equatorial horizontality flip to the southern version – and of course, vice versa? Most astronomical movements progress very slowly; unless we can visualize a slow and steady transformation between these different orientations as observer viewpoints change, I don’t think such solutions can be correct.

The description of the southern hemisphere analemma as being an inversion of the one we see in the north is clearly true. Viewed from the north, the smaller sector of the figure eight appears above the larger one, and in the southern hemisphere the smaller sector is lower down in the sky, and the larger one points upwards. It does indeed sound as though the shape in the sky must have reversed above the equator – but can that in fact be the case?

I want to suggest a different scenario for the transition from north to south. For simplicity I will only consider the analemma as plotted at solar noon, and will also focus on its vertical dimension, with very little to say about the width of its eight-like appearance except to acknowledge that this is divided into uneven sectors. Other lateral aspects of the shape are complex and do not seem needed for the discussion.

Imagine observers travelling longitudinally from north to south, the analemma is a vertical (almost) figure of eight with its larger sector lower in the southern sky. As they approach the tropic of Cancer, that analemma would remain vertical and move higher in the sky, until the tip of the smaller, upper loop – the summer solstice point – is directly overhead. As they progress further, into the equatorial zone, would they not expect to see that point move overhead behind them and the bottom of the larger, lower loop continue to rise? And when they reach the equator and then cross the tropic of Capricorn, the lower loop’s tip – the winter solstice point – would also pass directly overhead and begin to descend behind them. To see the analemma they would need to look up or lie on their backs and view the sky above their heads, in which case the larger loop would still appear to be below and the smaller loop (associated with the tropic Cancer and the summer solstice) would be higher -- at least with reference to their supine bodies.

But of course, they could remain standing and simply turn around, in which case their picture changes: the larger loop would now be higher in the (northern) sky and the smaller loop would be lower. But this change is caused by the observers’ about turn, and is no reason for thinking that the analemma has drastically changed in the equatorial zone. The larger loop continues to be the southern section of the analemma, but because it is now being viewed from the south it appears higher in the sky – and the smaller section of the loop, viewed from the south, appears lower because it represents the sun at points further north. (Turning around in this way also has other effects – for example, suddenly they perceive the sun as moving across the sky from right to left – though this is still east to west, of course.)

I hope this account paints a clear picture for you of my reasons for thinking the idea of horizontal analemma in the tropics is incorrect. Instead it should maintain its north-south orientation -- its apparent reversal in the southern sky is simply due to the fact that we are seeing it from a different perspective: down under. It is we who have turned around, not the analemma.

Extension #2 An attempted summary (Please note: Here I’ll lay out what I have gathered from the comments and answers posted on my original question. Though it’s a summary, it remains part of the question rather than an “Answer”, because I’d appreciate further feedback if people think the summary still contains mistaken ideas – or confirmation if you think that it is now essentially correct. Plus, I have additional analemma questions I would like to explore, and intend to post as separate questions later)

I see now that my original question, about internet sources stating that solar analemmas have a horizontal orientation if plotted at the equator, was misguided because I didn’t understand the way astronomers use the terms horizontal and vertical. By my (wrong) interpretation, the sources were saying that analemmas at the equator essentially underwent a radical change, switching from a north-south (“vertical”) orientation to an east-west (“horizontal”) direction. This idea was totally counter to my expectation, which was that they would remain north-south pointing from any location on the earth’s surface, not only in the temperate zones.

But of course, when discussing the sky as seen from a given point, astronomers determine celestial locations in terms of azimuth (degrees of horizontal separation as measured along the horizon from the first Aries point – the position of the dawn sun at the vernal equinox) and elevation (degrees of vertical separation from the horizon toward the overhead apex). Thus, a shape like an analemma is considered horizontal in whatever quarter of the sky it appears, so long as the two loops (or tips) are at the same elevation from the horizon. This of course includes north-south analemmas, which are horizontal if they are due east or due west of an observation point on the equator.

As it happens, to focus my question I selected the one type of analemma – the noon analemma – which, at the equator, centers on the actual apex of the sky, where it cannot be described as horizontal. Most equatorial analemmas, however, are based on the morning or the afternoon hours, and would be distributed along the celestial equator, where they would present more as infinity signs than figure eights. The observer would need look toward the east or the west to plot the sun’s positions for these analemmas.

Thus, the description of off-noon equatorial analemmas as horizontal infinity signs clearly does not contradict my expectation that their long dimension extends north-to-south (or vice versa) – rather, it confirms it. And an additional unstated expectation, that the morning-hour infinity-shaped analemmas will display their smaller loop to the right (pointing south) and that afternoon analemmas will show it on the left (also pointing south) remains plausible to me. I hope other readers will agree that this seems correct – or explain why.

There are other related quesiona that I would like to explore with respect to solar analemmas on earth, relating to the orientation of off-noon analemmas in the temperate zones of earth, and also the tip-to-tip length of analemmas at different latitudes. (Does this vary?) I plan to post these questions in the next week or two, and would greatly appreciate any feedback on them, especially from some of the contributors to this present chain of discussion, who have been very helpful!

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    $\begingroup$ See astronomy.stackexchange.com/q/54104/16685 $\endgroup$
    – PM 2Ring
    Jan 27 at 0:08
  • $\begingroup$ In simplest terms, what you're asking is why there's any lateral displacement from an imaginary straight line drawn between the max (summer solstice) and min (winter solstice) solar elevations at a given time over a year. I think this is a great question! I can understand why the Sun is higher or lower in the sky at (say) 3pm on a given day, but I can't figure why it's a bit east or west as well. I'm also intrigued to know what the "crossover" point in the figure-8 analemma represents, since it's not at an equinoctial point. $\endgroup$ Jan 27 at 4:48
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    $\begingroup$ Also related: how does the appearance of the analemma vary with latitude? $\endgroup$ Jan 27 at 22:12
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    $\begingroup$ Your extension is generally accurate and agrees with the answer from @DavidHammen. The analemma is a figure-8 regardless of where it is viewed from. The problem from the original question is we do not have the "several sources on the internet" that claim the analemma is (or should be) a horizontal line. Without that information, we cannot know what the websites are trying to say. As the answer indicates, one site says it would be a straight line on a different planet, but that does not mean a horizontal line. $\endgroup$
    – JohnHoltz
    Jan 31 at 1:34
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    $\begingroup$ Still curious at 82, it's important to remember that there are no parallel lines in spherical geometry. The lines of equal azimuth on the image provided by @PM2Ring are "lines" (a "line" connects two points via the path with the shortest distance). However, these lines intersect at the zenith and nadir points. The curves of equal elevation are not "lines". They are the intersection of parallel planes with the celestial sphere, but they're not lines. $\endgroup$ Feb 2 at 12:42

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The website timeanddate.com says this about the shape of the analemma:

If the Earth’s orbital path was elliptical, but its axis not tilted, the Solar Analemma curve would be oval shaped. At the equator, this line would be a straight line spanning from left to right or West to East.

However, the Earth rotation axis is tilted with respect to the Earth's orbit. At the equator, solar noon is to the north of vertical between the March and September equinoxes and to the south of vertical between the September and March equinoxes. The east-west component of the analemma essential is the equation of time, and this is due to both eccentricity and axial tilt. The north-south component (or up and down component at the equator is due to axial tilt. The analemma is not a straight line at the equator.

The analemma remains a figure eight at the equator. What does happen at the equator is that the axis of the figure eight becomes horizontal, so it is closer to the infinity symbol than a figure eight.

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  • $\begingroup$ With regard to the imgur visual, I’m unfamiliar with this source or what its parameters are, but isn't it showing an analemma seen from the equator located near the north pole?! I think you'd agree that an analemma is essentially a plot of the points in the sky from which the sun appears to shine on a particular location at a given hour each day of the year? The 11 AM analemma as seen on the equator would surely show the sun shining from nearly overhead throughout the year, but tracking to the north from the March equinox to the September equinox, and to the south for the rest of the year. $\endgroup$ Feb 1 at 4:10
  • $\begingroup$ @user23299429 Presumably you are writing about this image, posted by PM 2Ring in a comment to your question. That image depicts the position of the Sun in the the sky as seen at 11 AM from the equator. What you appear to be thinking might be the North Pole is the zenith: straight up. The Sun is slightly below the zenith and slightly to the east of the meridian (the line that divides east and west) as it is an hour before noon. $\endgroup$ Feb 1 at 11:52
  • $\begingroup$ Aah, thank you @David Hammen. This makes much better sense – but it also seems to show that the 11AM analemma is largely vertical, with its larger loop pointing north and its smaller loop reaching over equatorial line to the south – unless I’m still misunderstanding the parameters of image. (I’m assuming that for solar noon the analemma will be positioned higher, so that it lies along the meridian dividing east and west.) $\endgroup$ Feb 1 at 16:04
  • $\begingroup$ @user23299429 You are looking at that image incorrectly. The concentric circles centered about the zenith are curves of equal elevation, starting as a large circle at the horizon and shrinking to a point at the zenith, in steps of 15°. The curves that converge on the zenith are curves of equal azimuth, also in steps of 15°. A nearly vertical analemma would look very different from what is shown. This is nearly horizontal. The left and right ends both reach down to a bit above 60° elevation. $\endgroup$ Feb 1 at 18:23
  • $\begingroup$ I do believe that what I am finding difficult with your and PM 2RING’s explanations is simply the definitions of horizontal and vertical. I have used horizontal to mean parallel to the celestial equator, and vertical to mean perpendicular to the celestial equator, and thus parallel to the overhead north-south meridian. It has seemed more consistent to keep “vertical” for north-south orientation, because outside the tropics in each hemisphere the analemma does seem to have a largely vertical direction. $\endgroup$ Feb 2 at 0:39
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There is already a correct answer, but I think it could be complemented by drawing the analemma at the equator, where we can see that it's axis is horizontal:

analemma equator

Just for comparison, the analemma at Greenwich latitude:

analemma Greenwich

And near the North pole:

analemma north pole

The R code used to draw the annalemas, adapted from https://www.r-bloggers.com/2013/12/analemma-graphs/

library(oce)
loc1 <- list(lon=-0.0015, lat=0)  # Equator
loc2 <- list(lon=-0.0015, lat=51.4778)  # Greenwich Observatory
loc3 <- list(lon=-0.0015, lat=89.99)  # North pole
locs <- list(loc1, loc2, loc3)
t <- seq.POSIXt(as.POSIXct("2014-01-01 12:00:00", tz="UTC"),
                as.POSIXct("2015-01-01 12:00:00", tz="UTC"),
                by="1 h")
for (loc in locs) {
  sa <- sunAngle(t, lon=loc$lon, lat=loc$lat)
  par(mar=c(3, 3, 1, 1), mgp=c(2, 0.7, 0)) # tighten margins
  plot(sa$azimuth, sa$altitude, type='p', pch=20,
       xlab="Azimuth", ylab="Altitude")
  grid()
  abline(h=0)
}
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