why is the analemma a figure 8

Here is what I’m talking about. http://scienceblogs.com/startswithabang/2009/08/26/why-our-analemma-looks-like-a/
This particular website describes the answer…but…I’m not sure I understand it.
I understand that as the earth orbits the sun, the velocity will increase and decrease over the year, making the sun appear to lag behind or speed up relative to the average position. So with this in mind, I would expect the path to be a loop…not a figure 8. I still don’t get why its a figure 8. Instead of apparently speeding up then slowing down during the year, the figure 8 shows that the sun is lagging (march, feb) then speeds up (May, June); but then lags again (July, Aug), and speeds up again (Oct, Nov).
Why this apparent ‘slow, fast, slow, fast’ instead of just one ‘slow, fast’ cycle.

I think the answer is in that website…and I THINK it has to do with the idea that our clocks don’t exactly match the rotation of the earth, so we are artificially taking the measurement at the same clock time… not the same ‘earth’ time. But I’m not sure.
Can someone explain this to me using small words and simple thoughts. Am I correct that the “8” has to do with the interplay of our clocks not matching the earth rotation exactly?

No, the 8 has to do with the fact that the tilt of the Earth’s axis stays constant while the orbit is an ellipse.

I think I need more than that…

And (I think) with the fact that the solstices happen to occur very near the times when the Earth is closest and furthest from the Sun (Mars has tilt almost the same as Earth’s but it doesn’t have a figure-8 analemma).

Maybe this helps.


Yes, but why an 8 instead of a simple loop (as on Mars)? Because Earth happens to be closest to the Sun (therefore moving faster) during one of our solstices (specifically, the Southern Hemisphere summer).

(Edit: Ninja’d. This was in reply to Dr. Fidelius.)

A simple answer (see the nice graph is that the non-uniform motion of the sun in the sky can be regarded as a sum of two separate effects, one due to the eccentric orbit and one due to the tilt of the Earth’s axis with respect to its orbit.

These effects are roughly of the same order of magnitude and so combine to form an interesting curve. If the Earth’s orbit were more eccentric, like that of Mars, then you would see a simple egg-shaped loop.

I think we would get a symmetrical 8 if the solstices aligned with perihelion and aphelion.

If either the eccentricity of our orbit or our axial tilt were a little different we would not be seeing the loop from the “bottom” side for part of the year, which is why it traces a figure 8.

ETA: Mostly ninja’d. Since I typed it you can put up with reading it. :slight_smile:

Almost but not quite.

The East vs. West apparent position of the Sun in the sky is the sum of two things: the rotation of the Earth *and *the movement of the Earth around its orbit. The difference between 24 clock hours and the former is a constant. The latter effect changes over the year from a positive displacement from the average to a negative displacement and back.

So it’s sort of like a sine wave: Starting from a neutral position first the Earth speeds up around the orbit. Which means over time it “gets ahead” of the clock. More and more each day. Then the Earth starts to slow down. It’s still “ahead” of the clock at first, but now each day it’s lagging back towards being in sync with the clock. Eventually they line up again 6 months after the starting point. But now the Earth’s too “slow” compared to the clock. So it gets “behind the clock” and more and more each day. Eventually the Earth start speeding up again. And so begins to catch up to the clock. Eventually they line up again and we’ve completed one year.

If we had no axial tilt and you watched the analemma you’d see it first push out to the west, first accelerating the decelerating to a stop at the westernmost extreme. Then it’d accelerate back to the east, shoot through neutral and then slow back down to a stop at the easternmost extreme. Then accelerate back towards the west and back through neutral to complete one cycle.
Meanwhile North / South is much simpler. The Earth’s axis isn’t 90 degrees to the orbit. So the east/west sine wave described above is smeared in the north/south direction. If you take your analemma from the equator the two loops are a symmetrical figure 8. if you take your analemma from the Tropic of Cancer / Capricorn line there’s one big loop and one point-sized loop. Which means it looks like a simple oval / egg shape.

If you take your analemma from the mid latitudes, where most of us live, you get two loops of different North / South size.

Agreed (this happened in 1246), but I do not believe the date of the perihelion makes a difference to the figure-eight character of the analemma, i.e., over time the date of perihelion advances which shifts the curve, but it will still resemble an 8. Perhaps someone can confirm this.

At one point there was a program online that let you enter the eccentricity, tilt and timing of solstice relative to perihelion, and get an analemma as an output, but I can’t find it at the moment.

I do not believe these are correct statements.

Why is the Martian analemma more or less an ellipse? Its orbit is more elliptical than ours is. Is it that one of the lobes gets pinched to almost nothing or even unlaced entirely?

You can visualize the analemma without going to this extreme. In The Great Sundial Cutout Book the authors have pictures of images they’ve drawn on their walls by plotting the sun’s position as given by reflected spots on their walls at the same times of day, and they execute perfect analemmas, distorted by the geometry of the situation

The Great Sundial Cutout Book by Mablen Jones; Robert Adzema: Good Spiral-bound (1978) | Ergodebooks-used--PLA--v01&cm_mmc=msn--Bing+US+Shopping--Ad+group+%231--{product_groups}

Agree they’re not perfectly symmetrical. Because the line of aphelion/perihelion & the line of the nodes are different.

A more precise statement would have been the height of the two halves of the figure 8 will be equal (within a pretty small fudge factor) when seen from the equator. And the size of the poleward loop will be very small (within a pretty small fudge factor) at the tropic lines.

The OP wanted simple and with short words. There’s always a tradeoff in how much baby dandruff to send out with the bathwater.

THis is exactly what I have done in my office, which has prompted my question. I just now pretty much completed the pattern.

It looks more like an infinity symbol to me … perhaps as a metaphysical representation of God’s wisdom and might … or it could just be coincidence … everybody knows the torus is the divine shape of God’s wisdom and might … like a doughnut … yummy …

I think the point is that there is always a big loop and a little loop, no matter from where you observe. At the equator, the analemma is happening directly overhead, while near the poles part of it disappears below the horizon. Noplace does it look like on Mars or oddly distorted.

My main objection was to your statement that, at the Tropic of Cancer/Capricorn, the upper lobe of the analemma reduces to a dot. I don’t believe that is a correct statement.

Further, if I’m not totally mistaken, the difference in size between the north lobe and the south lobe has to do with the way the orbit is shaped and the fact that during a Northern Hemisphere winter, when the sun is low, the Earth is traveling faster than it does at Summer Solstice. So that would mean the lobes should be different even at the equator.

I would say… unlaced. It’s a smooth pear-shaped curve, not really elliptical at all. Here’s a picture.