Throwback Thursday: The Earth’s Analemma

What shape does the Sun trace out at the same time of day throughout the year?


“Life is tragic simply because the earth turns and the sun inexorably rises and sets, and one day, for each of us, the sun will go down for the last, last time.” -James A. Baldwin

Yet in just a few days, the Sun will reach solstice: the point at which Earth’s axis is maximally tilted with respect to the plane of our orbit. As our planet completes its annual revolution about our parent star, moving in a huge ellipse some 300 million kilometers along its major axis, it does so with a tilt. As our planet spins, it is neither parallel to nor perpendicular to our orbital plane, but rather tilted at a specific angle.

Image credit: Wikimedia Commons user Tauʻolunga, via http://en.wikipedia.org/wiki/Summer_solstice#mediaviewer/File:North_season.jpg.

When your half of the world is tilted towards our parent star, the path of the Sun through the sky appears longer, rises higher, and gives us more hours of daylight than average. This year, June 21st marks the Summer Solstice for the Northern Hemisphere, and the Winter Solstice for the Southern. As such, Europeans and North Americans will experience the longest day of the year this Saturday, while Australians will experience the shortest.

Image credit: Danilo Pivato, of http://www.danilopivato.com/.

In general, the Sun appears to rise in the Eastern portion of the sky, rise up high overhead towards the equatorial direction, and then lower down and set in the West.

If you were to track the position of the Sun throughout the year — such as through a pinhole camera — this is what you’d see.

Image credit: Kevin of Build it Solar, via http://www.builditsolar.com/Projects/Educational/Solargraphy/Solargraphy.htm.

The lowest and shortest path the Sun takes through the sky comes on the date of the Winter Solstice; the highest and longest takes place on the Summer Solstice.

Well, what would you expect, then, if you took a picture of the Sun — and its position in the sky — every single day at the same exact time for an entire year? Would you expect a straight line? A curved line?

I wouldn’t blame you if you did, but as it turns out, what you’ll actually see is much more surprising, and much more spectacular.

Image credit: Tunc and Cenk Tezel, via Astrosurf.

This shape looks like an elongated figure-8, with one lobe of the “8” much larger and wider than the other. The highest apex of this figure does, in fact, correspond to the Summer Solstice, and the lowest point corresponds to the Winter Solstice, as you’d expect.

But this shape isn’t a straight line at all, and it isn’t unique to our world. Uranus, Neptune and Pluto all see figure-8s from their world; Jupiter and Venus see ellipses; but Saturn and Mars see an unusual teardrop shape traced out through their skies!

Image credit: original source unknown, generated from Mars Science Laboratory with a little simulation help.

Why are these the shapes we see? And in particular, why is our analemma shaped the way it is and tilted the way it is?

The tilt is the easy part: if you took your photograph at high noon every day, you would get a completely vertical analemma: a figure-8 shape without a tilt at all.

Image credit: Hitoshi Nomura / Fergus McCaffrey, via http://fergusmccaffrey.com/artworks/analemma-91-noon/.

The analemma appears in this orientation — with the small loop at the top and the large loop at the bottom — because these images were taken from the Northern Hemisphere. From southern skies, the analemma appears inverted!

And, of course, the time-of-day makes a big difference in how this analemma is oriented: just as the Sun appears to move a little more quickly in the summer months across the sky, so does the analemma’s shape, and so analemmas taken at times other than noon appear tilted.

Image credit: Sydney Morning Herald, via http://www.smh.com.au/technology/sci-tech/column/astronomy/in-the-loop-yearround-20110616-1g48x.html.

So what accounts for this shape? Two things: our planet’s axial tilt, for one, and the ellipticity of our planet’s orbit as the other factor. If we didn’t have those: if the Earth orbited so that its axis was totally perpendicular to our orbital plane and our orbit made a perfect, uniform circle, the Sun would appear in the same place every single day. We would not only have no seasons, but its path through the sky would never change by a single degree from day-to-day, month-to-month or year-to-year!

But our planet’s orbit is tilted, and by a significant amount.

Image credit: Larry McNish / RASC Calgary Centre.

This 23.4° tilt means that the Sun’s path changes periodically throughout the year, rising higher and earlier in the Summer, lower and later in the Autumn, lower and earlier in the Winter, and higher and later in the Spring.

Our axial tilt changes a bit over time, but will remain relatively consistent over the next million years.

Image credit: Wikimedia Commons user Tfr000, via http://en.wikipedia.org/wiki/Axial_tilt#mediaviewer/File:Obliquity_berger_0_to_1000000.png.

If this were the only thing at play — meaning that our orbit was a perfect circle — our analemma would be a truly perfect figure-8, symmetric about both the horizontal and vertical axes.

But our orbit isn’t a perfect circle at all; it’s elliptical!

Image credit: Wikimedia Commons user Gothika, via http://commons.wikimedia.org/wiki/File:Seasons1.svg.

When the Earth is closest to the Sun, near but not coincident with the solstice, it moves its most rapidly near the Sun, causing the Sun’s apparent path to change more quickly than average. When the Earth is farthest from the Sun, it moves more slowly than average, causing the Sun’s path to change slower.

If we lived on an untilted planet that had an elliptical orbit, the Sun’s path through the sky would simply be an ellipse: this is what happens roughly on Jupiter and Venus, where the axial tilts are negligible. But here on Earth, we have both an elliptical orbit and a significant axial tilt, and so both effects are significant!

Image credit: Autodesk generated image via the UK.

On a planet like Mars, the orbit is extremely eccentric, and so the combined effects just create a giant teardrop, while on Earth, the fact that our planet moves fastest during the winter solstice makes the “lower” side of the analemma (from the Northern Hemisphere) so much larger than the “upper” side!

Image credit: US Coast and Geodetic Survey.

We like to think of each day as being 24 hours long, but in reality, the Earth makes a complete “spin” every 23 hours and 56 minutes; the remainder of our day comes from the Earth moving its position around the Sun. During the months nearest the June solstice (when the Earth nears aphelion, its farthest position from the Sun), it moves the most slowly, and that’s why this section of the analemma is “pinched.” But during the months nearest the December solstice (when Earth is near perihelion, closest to the Sun), it moves more quickly, and so that section of the analemma is elongated. (Mars, on the other hand, makes a teardrop shape because its perihelion and aphelion are lined up with its equinoxes rather than its solstices!)

So all told, we can combine these effects to make an equation for where the Sun will be located at any particular time: we call this the equation of time.

Image credit: Wikimedia Commons user Rob Cook, via http://en.wikipedia.org/wiki/Equation_of_time#mediaviewer/File:EquationofTimeandAnalemma.gif.

And this is what’s responsible for our analemma!

Image credit: György Soponyai of Budapest, Hungary, via http://spaceweathergallery.com/indiv_upload.php?upload_id=92546.

It’s the beautiful, natural shape the Sun traces out over time, and as our orbital axis precesses over time, our analemma will change in shape, too. So enjoy the Sun’s maximal highs (or lows) on the solstice, but remember what it’s doing throughout the year: making its cosmic pirouette through our skies!


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