APOD: Analemma 2010 (2010 Dec 31)

[neufer]

Please explain why the lower loop of the analemma is longer than the upper one. Since earth is at perihelion in January and moves faster then, the loop closer to the winter solstice should be shorter than the upper near the summer solsticeone. Earth is moving slower when at apehelion in August. This is going to bother me until someone explains it.

The sidereal rotation period of the earth is 23h 56m 4.1s but an extra 3m 55.9s is required (on average) to catch up with the apparent (59.14' long/day on average) longitudinal angular velocity of the sun around the earth.

However, the apparent longitudinal angular velocity of the sun around the earth is not a constant (54.23' long/day) for TWO different reasons.

Assuming the earth went around the sun in a circle then the sun would appear to go around the earth at a constant speed of 109.72km/day (40,075km/365.2422). However, at the equinoxes the ecliptic crosses the longitude lines at an angle of 23.5º so that the apparent longitudinal angular velocity of the sun is slowed down to ~54.23' long/day. At the solstices the ecliptic not only crosses the longitude lines directly (at zero angle) but the physical distant between longitude lines on the ecliptic is very short such that the apparent longitudinal angular velocity of the sun is speeded up to ~64.49' long/day. This explains the figure 8 pattern.

However, since the earth goes around the sun in a ellipse with perihelion around the northern winter solstice the apparent longitudinal angular velocity of the sun is especially fast [(54.23 + 12.43)' long/day] at this time whereas at aphelion around the northern summer solstice the apparent longitudinal angular velocity of the sun is only moderately fast [(54.23 + 8.16)' long/day]. The earth's slight elliptical orbit hence makes for an unsymmetrical figure 8. Mars's extreme elliptical orbit makes for a teardrop analemma. Pluto's extreme elliptical orbit is trumped by its extreme axial tilt and therefore has a figure 8 analemma.

[Guest]

Please explain why the lower loop of the analemma is longer than the upper one.

When the earth is moving faster in its orbit, the amount of time by which its position varies from that of a circular orbit is at its greatest. If the orbit was perfectly circular, the analemma would simply be a vertical line. It is the difference in speed during the ellipitical orbit that creates the lobes. The faster speed creates the larger lobe. It is explained in much greater detail (and mathematical precision) in the link I cited above.

[Boomer12k]

"TO INFINITY AND BEYOND!"

[alter-ego]

Please explain why the lower loop of the analemma is longer than the upper one.

When the earth is moving faster in its orbit, the amount of time by which its position varies from that of a circular orbit is at its greatest. If the orbit was perfectly circular, the analemma would simply be a vertical line. It is the difference in speed during the ellipitical orbit that creates the lobes. The faster speed creates the larger lobe. It is explained in much greater detail (and mathematical precision) in the link I cited above.

Your first statement is true, but deviation from a circular orbit / constant orbital velocty contributes only about 1/3 of the time variation magnitude. Less obvious, but significant, is the affect due to the tilted axis alone. This is likely created by Earth's rotation plane not parallel to orbital plane. For a circular orbit (eccentricity = 0) the figure 8 still exists, but is symmetrtic. Years ago I wrote an analemma program which cross-checks very well with the downloadable program available via the Other Analemmas tab. Below are interesting screenshots using the linked program:

Axis tilt explains about 2/3 of the time variation
(click to enlarge)

Only as the axis tilt => 0deg does the figure 8 converge to a point, but a simple line analemma does not ocurr. Unfortunately the linked program does not take inputs = 0, so if you want to explore really small tilts and eccentricities you can, but you'll get varied shapes.

[Beyond]

"TO INFINITY AND BEYOND!"

Why - Thank you, Boomer12k. YEE-HA!!

[Havanich]

I was wondering about yesterday's picture of the analemma viewed from central Europe. Does the shape of the analemma, specifically the altitude of the crossover, depend on location, time of day, or is it a constant? I thought the position of the crossover could depend on latitude, with the view at the equator being an ellipse. But perhaps not.

[quattroman]

I'm confused - how is the photographer's shadow in the picture? And doesn't that look more like a front yard then the backyard?

Because the pictures of the Sun used to create the analemma is photographed with a filter. To get one image of the scenery from the spot where the analemma pictures are taken, the photographer has to take one image with the sun out of the picture to avoid bakclight (in order to get a better picture).

[alter-ego]

I was wondering about yesterday's picture of the analemma viewed from central Europe. Does the shape of the analemma, specifically the altitude of the crossover, depend on location, time of day, or is it a constant? I thought the position of the crossover could depend on latitude, with the view at the equator being an ellipse. But perhaps not.

The Equation of Time does not depend on latitude - only on Earth's rotational and orbital parameters. The figure 8 shape likewise does not change with latitude, though parallax due to spatial separation (at differen latitudes) may change the exact RA/Declination coordinates of the Sun, the analemma shape and cross-over point (relative) location will not change.

However, it's not hard to see that latitude does affect the analemma's orientation, and that the orientation changes the Sun's rising and setting asymmetrically. For example, the analemma graphically shows why the Solstices aren't when the earliest/latest sunrises and sunsets occur even though they are the longest/shortes days. The plot shows a special case that I like - a rising analemma for at 12.5 deg latitude. Note the the Sun rises at the same time (within 10 seconds) over the entire month of September, while marching 15 degrees southward over the same time. For this case, the opposite side of the plotted analemma closely represents the slope on the western horizon. Over the same time interval, the sunset time changes by ~24 minutes.

Hope this helps.

[raindrop]

Thanks for explaining how the earth's axis tilt affects the analemna (sp?). At the same time you also explained why the sunset is earliest on Dec 8 and not Dec 20 or 21. I have been trying to figure this out for several years. I think the same causes go into moon rising time and why it rises closer together around the autumnal equinox. However, then you must be considering the moons tilt with respect to the earth? raindrop

[alter-ego]

Thanks for explaining how the earth's axis tilt affects the analemna (sp?). At the same time you also explained why the sunset is earliest on Dec 8 and not Dec 20 or 21. I have been trying to figure this out for several years.

Your welcome. There's more to the analemma than meets the eye, and it is nice when something abstract can tie into direct experience.

I think the same causes go into moon rising time and why it rises closer together around the autumnal equinox. However, then you must be considering the moons tilt with respect to the earth?

I'm confused what you are asking. A few things I'd like to say that might help answer / clarify your question. Bottom line, I believe your association of closer moonrise times to autumnal equinox is not generally correct.

First, the lunar orbit is very dynamic compared to the Earth's orbit. Moonrises and sets are changing very quickly compared to sunrises and sunsets. Whereas Earth orbital parameters change on the scale of 10's of thousands of years, lunar orbit precession / recession cycles have ~19year and ~9year periods. Generalizing a rise / set behavior as occurring at an equinox, or other specific time does not pan out. The elliptical lunar orbit does result in a nonuniform angular velocity, and this, in turn, results in rise / set time "bunching", i.e. times of the year when moonrises are close together, and times when they are far apart. The bunching occurs at different times in the year, and not necessarily the same times, given the changing lunar orbit. On average, the moon rises 51 minutes later each Earth day. If you are comfortable looking at number tables you can find data here and maybe additional data here.

Second, if you are relating the moon's rotation axis to the "moon's tilt", there is no connection to the moonrise / set times (at least wrt a "lunar" analemma as viewed from Earth). If you were on the moon observing the Earth, the tilt of the moon's rotation axis is a key factor for the Earth's analemma. As I stated, I'd expect this analemma would constantly change orientation and shape over time. Now, back on Earth, the angle of the lunar orbital plane with respect to the Earth's equatorial plane has a key influence in the moonrise / set times. However, a practical lunar analemma analogy comparable to the Sun's analemma does not likely exist because of the everchanging orbit. So thinking of moonrise and set times using an analemma model is difficult and impractical.

HOWEVER, you can generate an interesting "lunar" analemma snapshot (as seen in this APOD) by taking a picture of the moon (from the Earth) at time intervals increasing by 51 minutes each day to remove the mean lunar angular motion. This analemma reflects the higher-eccentricity lunar orbit by the significantly assymetric figure 8. Although the closer two adjacent moon images are in annlemma time (horizontal axis) also means that risetimes are closer, but remember that for this analemma, if images are parallel to the horizon then their risetimes are multiples of 51minutes apart (unlike the Sun's analemma where it would mean the same rise / set time).


I hope this is helpful.

[Havanich]

I was wondering about yesterday's picture of the analemma viewed from central Europe. Does the shape of the analemma, specifically the altitude of the crossover, depend on location, time of day, or is it a constant? I thought the position of the crossover could depend on latitude, with the view at the equator being an ellipse. But perhaps not.

The Equation of Time does not depend on latitude - only on Earth's rotational and orbital parameters. The figure 8 shape likewise does not change with latitude, though parallax due to spatial separation (at differen latitudes) may change the exact RA/Declination coordinates of the Sun, the analemma shape and cross-over point (relative) location will not change.

Rising Analemma, 12 deg Latitude.JPG

However, it's not hard to see that latitude does affect the analemma's orientation, and that the orientation changes the Sun's rising and setting asymmetrically. For example, the analemma graphically shows why the Solstices aren't when the earliest/latest sunrises and sunsets occur even though they are the longest/shortes days. The plot shows a special case that I like - a rising analemma for at 12.5 deg latitude. Note the the Sun rises at the same time (within 10 seconds) over the entire month of September, while marching 15 degrees southward over the same time. For this case, the opposite side of the plotted analemma closely represents the slope on the western horizon. Over the same time interval, the sunset time changes by ~24 minutes.

Hope this helps.

Thanks so much for your clear explanation. I hadn't thought about the time difference. APOD often shows us the complexity of our natural world. If the analemma moves toward vertical as you go north, would the bottom of the curve be below the horizon at the north pole?

[bystander]

Thanks so much for your clear explanation. I hadn't thought about the time difference. APOD often shows us the complexity of our natural world. If the analemma moves toward vertical as you go north, would the bottom of the curve be below the horizon at the north pole?

Yes, anywhere above the artic circle, the sun never rises in the winter and so the bottom of the curve would be below the horizon.