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#88597 - 12/03/02 08:42 PM analemma Part II, or, Days a-gettin' short...

Carpal Tunnel

Registered: 08/06/01

Posts: 5708

Loc: Worcester, MA more on the subject of the Analemma (perhaps more than we really want to know):

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Quoting from

Larry Denenberg <larry@intouchsys.com larry@harvard.edu> [thirteenth in a series] (12/96):

" The date of the earliest sunset varies with latitude. Today is the correct day for Boston. As you go farther north, the date gets later (up to about mid-December at 60 degrees north). As you go farther south, the date gets earlier---on the equator, it's in early November! Adjust accordingly.

It was pointed out to me that last year's explanation was misleading to say the least, or in other words totally wrong. Here is a more careful (and more correct) presentation based on an explanation from Andy Latto.

If you draw a line from the center of the sun to the center of the earth, that line touches exactly one point on the earth's surface. If you stand on that point the sun is directly overhead---that is, it is "solar noon" (although your watch probably doesn't say noon, because of our artificial time zones). It is also solar noon everywhere on the line of longitude going through this point, which is to say that the sun has risen as far as it's going to go. There are two things that make the sun-directly-overhead point move over time:

(1) The earth rotates on its axis. If you think of the earth not revolving around the sun and not tilted on its axis, it's obvious that the point moves along the equator, due west (since the earth turns west to east). Now think of the earth tilted in any way: the point still moves due west, along a parallel of latitude.

How fast does it move? Well, it moves entirely around the earth, 360 degrees of longitude, once per (sidereal) day, which is about fifteen degrees per hour. Since the earth turns at a constant rate, this speed is constant---as long as we measure in units of *degrees-of-longitude* per hour. The speed in *miles* per hour depends on which parallel of latitude we're following, but that doesn't matter: as far as the rotation of the earth is concerned, the sun-directly-overhead point moves due west, crossing lines of longitude at constant rate.

(2) The earth revolves around the sun. This makes the point move in a generally easterly direction, one revolution per year. To understand this motion of the sun-directly-overhead point, imagine the earth revolving around the sun, but not rotating---you'll see the point move around the earth once as the earth goes around the sun once.

Now, this 360-degrees-per-year motion would be due east if the earth's axis were perpendicular to the plane of the earth's orbit. But the earth's axis is tilted at 23.5 degrees. So the motion is along a great circle tilted at the same angle to the equator, tangent to the 23.5 degrees north and 23.5 degrees south parallels of latitude. The point is moving due east when it's tangent to these parallels. When the point is crossing the equator, it's moving either northeast or southeast, not due east. The point touches the parallels at the solstices (June and December) and crosses the equator at the equinoxes (March and September).

The point moves around this tilted circle at roughly a constant speed. But this means that the speed of the *easterly* component of its motion is not constant; it's largest when the point is moving due east (solstices), and is slower when the point is moving north or south as well as east. The speed is slowest when the point has the largest northerly or southerly component of motion, that is, at the equinoxes, when the point is crossing the equator.

[Digression: The point doesn't move around the tilted circle at a truly constant speed, because the earth's orbit is not a perfect circle---that's why I said "roughly" in the previous paragraph. The point moves slightly faster near perihelion (January) and slightly slower near aphelion (July). But the effect of this nonconstant speed is much less than the effect of the fact that the easterly speed varies depending on whether it's going due east or northeast. In particular, the easterly component of the motion is at a near-maximum in July, because the fact that it is moving due east is more important than the fact that it's going a bit slower than usual.]

Now we have to add together motions (1) and (2). (Remember, we're still discussing the motion of the point on the earth where the sun is directly overhead.) Motion (1) is due west, once around per day, and motion (2) is generally east, once around per year. Of course the sum is still basically west, basically once around per day, since motion (1) swamps motion (2).

But motion (1) crosses lines of longitude at constant speed, and motion (2) doesn't, so the combined motion doesn't! Near the solstices, the combined westerly motion is slower than average, since the easterly motion (2) is faster than average. Therefore, after the point crosses a given line of longitude, it takes more than the average 24 hours for the point to return to that line of longitude---that is, the solar-noon-to-solar-noon day is longer than average in December and July, and shorter than average in September and March. So the clock time of solar noon is a bit later every day near the solstice.

Since the length of the sunrise-to-sunset day is changing very slowly at the solstice, it also follows that the clock time of sunrise and sunset are both getting later at the solstice. But we know that the winter solstice is the shortest day, and that the time of the earliest sunset must be around the solstice somewhere. So if the earliest sunset is near the winter solstice, but the time of sunset is getting later *at* the solstice, then the earliest sunset must precede the winter solstice. Similarly, the latest sunrise must follow the winter solstice. We can also conclude that the latest *sunset* follows the summer solstice, and this is true---this year the latest sunset was on June 27 at latitude 40 degrees north.

Further discussion:

Another way to break up the motion of the sun-directly-overhead point is into two pieces as follows: One piece is the path the point would travel if the earth only rotated 365.25 times per solar year, and didn't revolve around the sun at all---this motion is similar to motion (1) above.

The other motion is the path the point would follow if the earth rotated exactly once per revolution, keeping the same face towards the sun, the way the moon does as it revolves around the earth. If the earth's orbit were a perfect circle, and the earth's axis were not tilted, the sun-directly-overhead point wouldn't move at all---the sun would always be directly overhead at the same point on earth (no doubt somewhere in Omaha, the tanning capital of the world).

Still considering the second motion, the effect of the eccentricity of the earth's orbit is to make the point oscillate slightly, once per year, moving east in the Northern hemisphere winter, and west in the Northern hemisphere summer. The effect of the tilt of the earth's axis is to make the point move in a figure eight, moving east at the solstices, northwest at the spring equinox, and southwest at the fall equinox. The path it follows, as a result of these two effects, is called theanalemma. Since the effect of the tilt is larger than the effect of the eccentricity of the earth's orbit, the general shape is a figure eight. The effect of the eccentricity is to squash the figure eight slightly, making the northern hemisphere lobe slightly smaller. It's also slightly asymmetric, because the perihelion is not exactly at the same time as the solstice. "

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