# Why would the Earth's solar analemma would be still a figue eight even if Earth's orbit was circular?

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The Earth's solar analemma is a diagram showing the deviation of the Sun from its mean motion in the sky, as viewed from a fixed location on the Earth… For [a planet] with a circular orbit but significant axial tilt, the analemma would be a figure of eight with northern and southern lobes equal in size.

This statement seems to be true, but it doesn't make any sense to me. If the eight is not due to the equation of time, why is there an eight? Thanks.

It is correct that the Earth's axial tilt alone would result in an analemma of a figure of eight with equally sized lobes. The axial tilt contributes to the equation of time.

Circular Orbit, no Axial Tilt

Let's look at an example case with a perfectly circular orbit and no axial tilt, to observe what happens with no equation of time. In this example, it's noon at the equator and prime meridian. The Sun is directly overhead. Next, the Earth rotates exactly once. This isn't a normal solar day, but a sidereal day, about 23 hours, 56 minutes. The Earth has rotated exactly once with respect to the stars. Where is the Sun directly overhead now? The Earth has moved a little bit in it's orbit around the Sun, so that our observer at the equator and 0 degrees longitude has observed that the Sun has moved in the sky with respect to the stars. The Sun is now directly overhead at a location to the east, a little less than 1 degree longitude away (360 degrees in a circle, usually 365 days in a year). The Earth must rotate a little more so that the Sun is directly overhead again (which completes the 24 hours in a day).

After each rotation, the Earth must rotate a little more each day to bring the Sun back overhead at 0 degrees latitude, 0 degrees longitude. After a quarter of a year, the Earth must rotate another quarter of a rotation (90 degrees) for the Sun to be overhead at that spot. Without that extra quarter of a rotation, the Sun is overhead at 90 degrees east longitude. That makes sense; the Earth has moved 90 degrees in its orbit. With a perfectly circular orbit, the rate at which this point moves eastward per day is constant, a little under a degree per day.

These points define a great circle around the Earth that coincides with the equator. A great circle is a circle on the surface of a sphere whose center is also the center of the sphere; it's the biggest possible circle that can exist on the surface of a sphere.

A Large Axial Tilt

Let's keep Earth's perfectly circular orbit, but let's give it a very large axial tilt, 80 degrees, for the purpose of this explanation. This Earth now begins at noon on the northward equinox at 0 degrees latitude, 0 degrees longitude. After one Earth rotation, where on Earth is the Sun overhead now? It's still a little less than one degree of angle away, but the direction of the displacement has changed. Instead of this point moving due east, it is mostly northward and only a little bit east. The Earth needs to rotate far less than 4 more minutes for the Sun to reach local noon for our observer at 0 degrees latitude, 0 degrees longitude. These points are west of where they would be without axial tilt. Solar noon has occurred before 24 hours have passed. As days continue to pass after the equinox, solar noons continue to occur earlier each solar day, as these points continue mostly northward and only a little bit eastward. The analemma shape as seen in the Northern hemisphere sky is moving up and to the right (west is to the right when looking south towards the Sun in the Northern hemisphere).

A quarter of a year later, it's the northern solstice, so let's see what has happened to our sun-overhead plots as the sidereal days have piled up. These points have continued to move mostly northward and a little eastward, but now they are moving purely eastward, because at the solstice the sun has stopped moving northward in the sky. It's still moving at a little less than 1 degree per day along that great circle, but because longitude lines are spaced much more closely together at 80 degrees latitude, one degree along a great circle covers many degrees of longitude. In other words, the eastward movement of these points has been "catching up" with the longitude of the corresponding no-axial tilt points. At the solstice, both the no-axial tilt point and the axial tilt point have the same longitude, 90 degrees east. The analemma shape as seen in the Northern hemisphere sky is moving up and to the left (eastward), where it reaches its highest point.

These points will continue to move through longitude lines rapidly while the Sun's overhead point remains far in the north. Now, the sun's overhead point is further east than it would be with no axial tilt. The analemma shape as seen in the Northern hemisphere sky is moving down and to the left (eastward).

As the southward equinox approaches, the sun-overhead points are moving mostly southward and only a little eastward. The analemma shape as seen in the Northern hemisphere sky is moving down and to the right (westward). This allows the non-axial tilt points to "catch up", and the longitude differences begin to decrease again, until the time of the southward equinox comes, when the two points coincide again.

Here is a crude ASCII drawing of these points so far, with longitude along the horizontal direction, and latitude along the vertical direction.

(2) K L M N O J P I Q H R G S (1)F T(3) E U D V C W B X A B C D E F G H I J K L M N O P Q R S T U V W X Y A = northward equinox M = northern solstice Y = southward equinox

The points around (1) are further west than their corresponding points on the equator. At (2), the points have caught up in longitude. At (3), they are further east than their corresponding points on the equator, but the equator points are catching up.

After the southward equinox, the longitudes of the sun-overhead point diverge once again, with the same mechanism as described above for the northward equinox. Just reverse north and south, and the eastward movements are the same. The analemma shape as seen in the Northern hemisphere sky first moves down and to the right, then down and to the left where it reaches its lowest point at the southern solstice, then up and to the left, then up and to the right where it reaches its starting point at the northward equinox.

Back to Reality

With an axial tilt of 80 degrees, the equation of time would show some extreme values, approaching almost 6 hours of divergence from the no-axial tilt case. With the true axial tilt of about 23.5 degrees, our equation of time difference values are far less substantial, but the effect is real.

The true analemma shape we see on Earth is the combined effect of the axial tilt that we have seen plus the fact that Earth's orbit is elliptical and it slows down during the parts of its orbit when it's farther from the Sun and it speeds up during the parts of its orbit when it's closer to the Sun.

The site Analemma has good explanations about the individual effects of the elliptical orbit and the axial tilt and how they combine to create our analemma.

## On the Trail of the Analemma

Canadian potter Steve Irvine took this photograph of the analemma over a monument known as Keppel Henge in Ontario, Canada. The image of the monument was taken as the final photograph in the series, on the same frame as the analemma series. Courtesy of Steve Irvine

If you want to photograph the analemma -- the yearlong pattern the sun etches in the sky as the Earth orbits around it -- there are lots of things to worry about.

Forty-year rainstorms, unstable cameras and balky batteries -- these are just a few. Ask Anthony Ayiomamitis, who has encountered all of it.

Fortunately for Ayiomamitis, he walked away from the experience with nine complete photographs of the phenomenon -- well more than what he needed to become the eighth known person in the world to have ever photographed a perfect solar analemma.

Ayiomamitis, a retired Greek systems design consultant who has made astrophotography his full-time job, and the seven others managed to capture on film the figure-eight pattern that the sun makes as its position in the sky slowly shifts over the course of a year. The phenomenon is the result of the Earth's tilted axis and its elliptical orbit around the sun. And though astronomers have long known about the pattern, it can only be seen by taking a photograph of the sun at the same time every day for an entire year.

The first such photograph -- or rather, series of photographs all on one frame of film -- was taken by Sky & Telescope magazine editor Dennis Di Cicco between 1978 and 1979. The result is a luminescent figure-eight in the sky, with one loop slightly larger than the other.

Measuring the height of the figure can help amateur astronomers confirm that the Earth is indeed tilted on its axis by 23.45 degrees. Measuring the width at different points reveals how the Earth travels past the sun at different speeds depending on the time of the year.

"(An analemma photograph) is certainly something that the scientific community regards very highly, for the camera is able to capture something passive and subtle which our eyes cannot appreciate from day to day, and certainly not over the course of a year," said Ayiomamitis. "It documents the dynamics of the universe and how changes, slow or large, can be explained using scientific observation and logic."

What makes capturing the analemma on film so difficult has more to do with photography than with astronomy. Photographers must commit to returning to the same location every several days over the course of an entire year. And because all the shots must be taken on the same film frame, any one mistake can ruin all the previous shots, no matter how perfect they were.

Also, the photographers must ensure that they can set the camera up in the exact same spot on each visit or mount it permanently at the selected location. And they must press the shutter within seconds of the scheduled time, or else that particular image of the sun will be out of line with the others.

Most of the time, errors in the process are not discovered until the photographer finishes the project after a year and takes his film into the darkroom to be processed.

"The problem is that there are many things that can go wrong and a single instance of Murphy's law during the yearlong marathon is sufficient to kill an effort to capture this phenomenon," said Ayiomamitis. "If the film moves within the camera chamber, we have a wasted effort. If the camera is not oriented perfectly and the analemma at some point during the year eludes the field of view, we have a blown effort. If the camera fails -- like the electronics, the battery, etc. -- we have a wasted effort."

Even if a photographer manages to control all these variables, weather is still a factor to worry about. Waiting a week or more for clouds to disappear can leave gaps and ruin an otherwise perfect analemma pattern.

"For that year, I was watching weather reports carefully," said Steve Irvine, a potter and amateur astronomer who photographed the analemma from a farm in Canada. "Two-day forecasts are OK, but five-day forecasts are hit-or-miss. I ended up deciding that if the sun was out on the day before my scheduled day I would just take the picture then. It was kind of like bird-in-the-hand thinking."

Another big obstacle was the temptation to skip a photo opportunity, especially on mornings when the ground was covered in several feet of snow, said Irvine.

"After doing it for a couple months, you can get a little complacent," he said. "Even after taking your 30th image, you have to stay focused."

But with all the inherent challenges, the photographers didn't hesitate to throw in a few of their own. For instance, Irvine captured the image of the foreground scenery for his analemma shot on the same frame as his images of the sun -- a risky maneuver that could have ruined the frame, had it been underexposed or overexposed. (The foreground is generally blocked out with a filter when taking images of the sun so that it is not overexposed or blurry on the resulting photograph.)

Ayiomamitis, on the other hand, avoided that problem by taking the foreground images on a separate frame, after he had taken the images of the sun, and then using Photoshop to bring the two together.

He did, however, take it upon himself to photograph the analemma at not just one time of the day, but rather at 11 different times as part of a set that shows the position of the sun changing from sunrise to sunset. An additional image shows two analemmas on one frame of film.

Making it a point to begin each photograph in January and end the series in December, Aviomamitis effectively captured the full analemma in one calendar year. Photographers in the past have started in the middle of a year, causing them to finish the series in the following year.

In Athens, Ayiomamitis successfully photographed the analemma at eight different times of the day so far for his full-day series. He has three more to go before he completes his goal. If he fails, he'll have to wait until the following January to begin again, by his own rules.

Despite the difficulty of it all, he encourages others to give it a try -- especially from regions of the globe where the analemma has not yet been photographed. Because each region has a different view of the sun, the photographs will be unique.

"We do not have any examples of the analemma from the southern hemisphere, where the analemma would be inverted, where the larger loop is up top and the smaller loop is at the bottom," said Ayiomamitis.

"Similarly," he added, "we do not have any examples of the analemma from someone around the equator where the early morning and late-afternoon analemma would be perfectly horizontal." Such a photograph would show one half of the analemma above the horizon while the other half would be hidden below it.

"To this end, this work would be a great motivator for someone," said Ayiomamitis. "However, it should be emphasized that this is a very difficult challenge which requires huge amounts of patience, perseverance and advance planning and discipline, not to mention some good luck."

## Astronomy Final Exam

A) It doesnʹt. Earthʹs orbital distance plays no significant role in the seasons.

B) It causes the seasons to be more extreme than they would be if the Earthʹs distance from the Sun were always the same.

C) It is responsible for the fact that the seasons are opposite in the Northern and Southern hemispheres.

A) they were the first people to realize that Earth is a planet orbiting the Sun

B) the books of every other culture were lost in the destruction of the library of Alexandria

C) they were the first people known to try to explain nature with models based on reason and
mathematics, without resort to the supernatural

A) The electron jumps to level 3 as soon as it absorbs any additional energy.

B) A different electron drops into level 1, since it is now unoccupied.

C) The electron remains in level 2 until it absorbs an additional 10.2 eV of energy.

A) Erosion affects the maria more than it affects other regions of the Moon.

B) The maria formed after the heavy bombardment ended.

C) The maria formed within the past 1 billion years.

A) metals condensed first in the solar nebula and the rocks then accreted around them.

B) metals sank to the center during a time when the interiors were molten throughout.

C) radioactivity created metals in the core from the decay of uranium.

A) The large gap known as the Cassini Division is shaped by an orbital resonance with the moon Mimas,
which orbits well outside the rings.

B) The rings are so thin that they essentially disappear from view when seen edge‐on.

C) The rings must look much the same today as they did shortly after Saturn formed.

A) Individual ring particles orbit their planet in accord with Keplerʹs laws, so that particles closer in orbit
faster than particles farther out.

B) Saturnʹs rings formed along with its moons 4.6 billion years ago.

C) All four jovian planets have rings.

A) There would be a large empty region in our solar system between the orbit of Mars and the Kuiper belt.

B) Neither the asteroid belt nor Oort cloud would exist.

C) Earth would have suffered far fewer impacts.

A) Earth would be inside the supergiant.

B) Earth would fly off into interstellar space.

C) The supergiant would appear as large as the full Moon in our sky.

A) G stars are too cool to excite most hydrogen atoms to the first energy level, from which they can then
absorb visible wavelengths of light.

B) At these high temperatures, nearly all the hydrogen is ionized, and unable to interact with light.

A) It is the edge of the black hole, where one could leave the observable universe.

B) It is the ʺpoint of no returnʺ of the black hole anything closer than this point will not be able to escape
the gravitational force of the black hole.

C) The term is intended to emphasize the fact that an object can become a black hole only once, and a black
hole cannot evolve into anything else.

A) There are no white dwarf binaries to trigger star formation via their supernovae.

B) All of the galaxyʹs cool gas settles to the galactic plane.

C) The halo has no gas at all.

A) gravitational potential energy released by matter that is falling toward the black hole

B) jets emerging from the accretion disk

C) matter‐antimatter annihilation occurring just outside the event horizon of the black hole

A) most civilizations destroy themselves within just a few hundred years of arising

B) primitive life is common but intelligent life is rare

C) most of the civilizations that have ever existed are still out and about in the galaxy

A) They only determine relative ages by assuming that deeper layers of sedimentary rock are older.

B) They compare the ratio of carbon‐12 to carbon‐13 in the fossil to determine how long ago it formed.

C) They use radiometric dating to measure the age of the surrounding rocks.

A) The average distance between galaxies is increasing with time.

B) The statement is not meant to be literal rather, it means that our knowledge of the universe is growing.

A) The technique of stellar parallax was used by Hubble to determine that the Andromeda Galaxy (M 31)
is about 2 million light‐years away.

B) Ancient astronomers were unable to measure parallax and used the absence of any changes in the starsʹ
separations as an argument in favor of an Earth‐centered universe.

C) You can demonstrate parallax simply by holding up a finger and looking at it alternately from your left
and right eyes.

D) The existence of stellar parallax is direct proof that Earth orbits the Sun.

## Why would the Earth's solar analemma would be still a figue eight even if Earth's orbit was circular? - Astronomy

by Pat Murphy & Paul Doherty

Gordon Van Gelder, the editor of this magazine, is a clever man. When he asked

us to write for Fantasy & Science Fiction, he wanted us to write four columns a

year. He cleverly set our deadlines on dates that we, as scientists who spend our

time observing the natural world, could not possibly forget. He said, "Let's

make your columns due on the solstices and equinoxes."

Very clever. We haven't forgotten a deadline yet.

In honor of that astute decision, we decided to write this column about about the

solstices and about the movement of the sun as perceived from the earth. We're

going to suggest that you spend some time watching shadows, a way of

indirectly observing the movement of the sun across the sky. These observations

can put you in touch with natural patterns that humans have been watching for

thousands of years--but that most of us modern folks have come to ignore.

Along the way, we'll talk about time--we just can't avoid it.

I'm not lazy I'm a scientist

We'll start with an observation you can make on a sunny afternoon, while

lounging around in a hammock or kicking back in a poolside bar. Take a look at

building or a fence, makes a straight line. Mark that line somehow--with a rock

if you're on the grass, a chalk line if you're on blacktop, or a swizzle stick if

you are sipping daquiris by the pool.

Guess where the shadow will be in 15 minutes and mark your guess in the same

way you marked the shadow. ("Hey, could we have some more swizzle sticks

over here? Oh, sure--put 'em in another round of daquiris.")

In 15 minutes, check your guess. You may be surprised at how quickly the

shadow moved. That is, you may be surprised at how quickly the sun moved.

Or, to be even more accurate, you may be surprised at how quickly the earth is

spinning--about 1000 miles per hour at the equator.

Right now, for simplicity's sake, we're going to talk about the movement of the

sun. We know and you know (and Galileo knew) that the sun isn't really moving

across the sky. But according to Paul, physicists have to be adept at jumping

from one point of view to another. According to Pat, so do writers. So we're

going to stay on earth (at that poolside bar, maybe) and tell our story from that

frame of reference for a while.

Next time you spend the day outside, pay attention to the movement of the

shadows as they move with the sun. The sun rises more or less in the east (more

on that "more or less" later) and sets more or less in the west. So the shadows

point more or less west in the morning and more or less east in the afternoon.

An interesting aside here: if you watch the movement of the shadow on a sundial

over the course of a day, you'll notice that it moves in a clockwise direction.

Coincidence? We don't think so! Early clockmakers designed clocks to mimic

the familiar sundial. Those readers who are fond of alternate history stories

might consider what might have happened if we used clocks based on sundials

that had been developed in the Southern Hemisphere where shadows move the

Shadows are at their longest at sunrise and sunset. When are they at their

shortest? Noon, you say? Well . . . more or less. You see, unlike sunrise and

sunset, the concept of noon relates to human time-keeping--and that gets a little

Does Anybody Really Know What Time It Is?

If you are telling time by the sun, noon is defined as the time when the sun is at

its highest point in the sky. The important words in that sentence are "if you are

telling time by the sun." That is, if you are using solar time. Chances are, you

are telling time by that device strapped to your wrist. And the time on your

wrist watch isn't solar time it's what's called standard time. You can blame that

Back before 1883, people used solar time. Each community kept its own time,

basing that time on the sun's position in the sky. Since the sun is always moving

across the sky, noon where you are is at a slightly different time than noon at a

place a few miles to the east or west. Back before 1883, noon in one town would

be four minutes later than noon in a town fifty miles to the east.

In 1883, to regulate time for the sake of railroad schedules, the United States

adopted standard time, designating time zones and requiring all communities

within a time zone to keep the same time&emdasheven though that standard time

doesn't quite match solar time.

If you are smack dab in the middle of your time zone, the sun will be at its

highest point at noon. But if you are at one edge of your time zone, solar time

may differ from standard time by as much as 40 minutes.

If you spend some time watching shadows, you'll notice that the position and

length of a shadow depend not only on time of day--but also on the time of year.

That's because the sun's position at a certain time is different in different

seasons. And that, of course, brings us to the solstices.

What's the longest day of the year? Any good Druid could tell you the answer to

that one. The longest day is the summer solstice (June 21 or thereabouts) and the

shortest day is the winter solstice (December 21 or thereabouts).

As a knowledgeable fantasy reader, you probably even know of some of the

fantasy connections for these dates. The summer solstice is associated with

Midsummer's Night Eve, when witches and fairies and other supernatural forces

are in control. In The Hobbit, the keyhole that lets Bilbo, Thorin, and the

dwarves unlock the passage into the dragon's lair opens on Durin's Day, the first

day of the last moon of autumn on the threshold of winter.

You know the length of a day changes over the course of a year, but have you

ever really paid much attention to the position of the sun--other than squinting

when the summer sun comes in your window too early or complaining when the

days get too short? Well, here's your chance.

For those of you with a lot of patience, here's an activity that takes a year to

complete. You need a south-facing window, a pocket mirror, some small

Post-Its, and a lot of patience. Choose a time of day when you'll be home at least

once every couple of weeks for the next year. Put your little mirror on the

window sill and position it so that it reflects a spot of sunlight on the wall or the

ceiling. Cover most of the mirror with masking tape, leaving only a 1/4" square

If you can, fasten the mirror in place so no one moves it by accident. Pat stuck

hers down with some stuff called "museum putty," that's sold in California

under the brand name Quake Hold™. You folks who are sensible enough to live

far away from the fault zone will have to come up with your own methods.

Note the time and date on a Post-It, and stick the Post-It to the wall or ceiling

where the spot of light reflecting from your mirror falls. A week later, at the

same time, do it again. And a week later, do it again. Repeat for an entire year.

(We warned you that you'd need patience.)

As you do this, you need to use standard time. If you move your clock forward

(or back) to adjust for daylight savings time, change the time that you make

your weekly mark by an hour.

Keep it up, and at the end of a year, the Post-Its will form a figure eight on

your wall. The marker from mid-December will be at one end of the eight and

the marker from mid-June will be at the other. This pattern is called the

analemma, which is Latin for "sundial." The analemma is a visual record of the

sun's changing position over the course of a year.

This is the same figure 8 you see on earth globes&emdash usually in the middle of the

Pacific Ocean. It is also known as the equation of time. Each planet has its own

shape for the analemma. On Mars, the analemma is the shape of a teardrop.

Pat is in the middle of doing this activity. (On earth, not on Mars.) As we write

this column, she has Post-Its all over the ceiling of her sunporch. She started

back in December and we're writing this in March, so she's not even halfway

If you (like most of us), prefer instant gratification, then you probably have

access to the World Wide Web. In that case, we suggest you visit

www.skypub.com/spc/staff/dic.html. On that web site, you find Dennis di

Cicco's award- winning, year-long photograph of the analemma made in the late

1970s. But to convince yourself that Dennis didn't cheat and do this in a

darkroom, you still might want to try the experiment with a mirror and

Post-Its. Depends on how trusting you are. (According to Paul, scientists must

be professional doubters. But he's not the one with Post-Its all over his ceiling,

You want to know why the analemma is a figure-eight, rather than a tear-drop

or an oval or a circle? You fool! Pat wanted to know why, once upon a time.

Days later, after much explanation with circles and arrows and too many

diagrams and too much math, she decided she didn't want to know the whole

We're going to give you the short version of why the analemma is a

figure-eight. If you must understand every last detail (which Pat claims is

enough to make a person's head explode), we recommend you visit

www.analemma.com, a thoroughly detailed Web site with animations and full

discussion of why the sun does that.

We'll start you off with an easy question: where does the sun rise? Did we hear

you say "east"? Sorry. It's an easy question, but the answer is tricky. We warned

If you were to watch the sun rise each morning over the course of an entire

year, you'd see that the sun doesn't always rise in the same place. In the

summer, in the Northern Hemisphere, the sun rises a little bit north of due east.

The date on which it rises the farthest north of due east is June 21, the summer

solstice and the longest day of the year. In the winter, in the Northern

Hemisphere, the sun rises a little bit south of due east. The date on which it rises

the farthest to the south is December 21, the winter solstice and the shortest day

Suppose you watched the path of the sun on the winter solstice and on the

summer solstice. On the summer solstice, the sun rises much higher above the

horizon at noon than it does on the winter solstice, taking a longer path across

the sky. On the winter solstice, the sun never gets as high in the sky.

Okay, now we're going to have to do one of those shifts in viewpoint that

physicists and writers like. Instead of staying on earth, we need to take a look at

the solar system from the outside, examining the earth's orbit.

The sun's path across the sky changes with the seasons partly because the earth's

axis (the imaginary line through the earth around which the planet spins) is

tilted with respect to the earth's orbit around the sun. As the earth orbits the

sun, the North Pole (the point where the axis intersects with the earth's

Northern Hemisphere) always points in the same direction, pointing near

Polaris, the North Star. (The direction of the Earth's axis does change over a

26,000 year cycle, which means that the analemma evolves with time. But we're

not going to get into that here.)

Because the earth's axis is tilted, during a portion of the earth's orbit, the earth's

Northern Hemisphere is tipped toward the sun. That's when it's summer in the

Northern Hemisphere. The North Pole is tipped toward the sun and the sun

shines on a greater area of the Northern Hemisphere. As the earth spins, places in

the Northern Hemisphere stay in the sunlit area longer, and the days are longer.

At the other extreme of the earth's orbit, the earth's Northern Hemisphere is

tipped away from the sun. That's when it's winter in the Northern Hemisphere.

The sun shines on a smaller area of the Northern Hemisphere, and the days are

The earth's tilt affects the position of the sun in the sky&emdashand so does the shape

of the earth's orbit around the sun. You might think that the earth always

traveled about the same speed on its way around the sun. That would be the case

if the earth's orbit were circular&emdashbut it's not. The earth's orbit is an elliptical,

which means that sometimes the sun is closer to the earth and sometimes it's

farther away. The difference in distance is only about 3% of the overall

distance. That may not seem like much, but it makes a difference to the speed of

Suppose you took the average speed of the earth --about 30 kilometers per

second. If you checked the planet's speed when it was closer to the sun (which

happens in January), you'd find it was a little faster than that average. When the

earth was farthest from the sun, in July, you'd find that it was moving a little

That's what all this looks like from outside the solar system. How does all this

affect what you see on the planet Earth?

Paul says that the analemma would be easier to understand if there were no

atmosphere on the earth. Without the atmosphere to scatter the sun's light, we

could see the stars during the daytime, and we'd be able to see the sun's

movement against the background stars. (Of course, if there were no

atmosphere we couldn't breathe. But we'd understand the analemma. Pat says

that seems like a small consolation.) Anyway, if we could see the sun moving

against a background of stars, we'd see that the sun moves on a regular path

through the stars, a path called the ecliptic.

Suppose you could see the stars when the sun is out. Suppose you're watching

the stars at around noon in mid-May. The sun is in the constellation of Scorpio,

perhaps near the star called Antares. Just before noon the next day, 23 hours and

56 minutes later, you check on the position of Antares. Antares will be back in

the same place in the sky. The sun, however, won't yet have reached its highest

point in the sky. That will take about four more minutes. From your point of

view on earth, the sun is lagging four minutes behind the stars.

the four minutes that it takes to get the sun back to its original position. You get

24 hours or one average day. Very tidy, isn't it?

Each day, the sun lags behind the stars. Over the course of months, this

accumulating difference means that different stars rise at different times. In the

Northern Hemisphere, for example, Scorpio is a summer constellation--you

don't see it in the night sky during the winter. The difference between the sun's

movement and the stars' movement has shifted the rising time for the stars that

make up Scorpio so that the constellation is up during daylight hours.

Why do the sun and Antares move across the sky at slightly different rates? Ah,

that takes us back to outer space. The earth is spinning, and that's what brings

Antares back to its starting position. But the earth is also orbiting the sun. That's

why it takes an extra four minutes (or so) for the sun to get back into position.

No doubt you caught that weasely little "or so" in the previous sentence. It

doesn't always take the sun exactly four minutes to get back into place. After all,

the earth isn't always orbiting the sun at the same speed. From your point of

view on earth, that means that the time it takes the sun to return to a particular

place in the sky isn't always the same. In early January, when the earth is nearest

to the sun, the sun moves farther from one day to the next. It takes longer for

the earth to overtake the sun and return it to the same place in the sky. It can

take 8 seconds longer each day. These 8 seconds add up from day to day, and the

sun begins to lag behind. In June, when the sun is farthest from the earth, the

sun takes less than four minutes to return to its original position.

But that's not all. Remember the earth's tilt? Over the course of the year, the

position of the noontime sun moves up and down in the sky, because of the

earth's tilt. Consider the position of the sun at noon. Maybe you've been told

that the sun is overhead at noon. That's not necessarily so. (Sorry. Someone's

been telling you fibs.) In fact, if you are in North America, the sun is never

directly overhead. For the sun to be directly overhead, you have to be in the

tropics, the belt around the earth between the Tropic of Cancer at 23.5 degrees

north latitude and the Tropic of Capricorn at 23.5 degrees south latitude.

On the summer solstice, when the North Pole is tilted 23º 21' toward the sun,

the sun is directly overhead on the Tropic of Cancer. Six months later, on the

winter solstice, the South Pole is tilted toward the sun and the sun is directly

overhead on the Tropic of Capricorn.

From our earth-based viewpoint, the movement of the sun changes in two ways

over the course of the year. The daily path of the sun moves up and down in the

sky, and the time it takes the sun to reach its noontime position changes, with the

average time being four minutes.

Those of you who are familiar with electronics may have seen Lissajous figures,

very cool patterns that appear on an oscilloscope screen when you have two

signals out of phase with each other. Paul says of the analemma: "It's a Lissajous

figure with the sun moving up and down in the sky once a year and ahead of and

behind the rotation of the earth twice a year." Pat agrees that makes a certain

amount of sense: after all, you have two movements that are out of phase and

that could certainly create a figure 8. But she says that we have already caused

the heads of our audience to explode and we should stop now.

For those of you who are still with us, here is one more question. You know

that the winter solstice is the shortest day of the year. On what day of the year

does the sun rise latest? Or, for those of us who prefer not to be up at dawn, on

what day of the year does the sun set earliest?

Did you say the winter solstice? Not a bad guess, but wrong, nevertheless.

Though the winter solstice is shortest day, but it's not the day when the sun rises

latest or sets earliest. The exact date of the latest sunset depends on your exact

latitude, but around here the earliest sunset is around about December 7 or so.

The latest sunrise is around about January 4. And the winter solstice is

December 21, somewhere between the two.

Weird. To understand why this happens, you need to apply the concept of

analemma rise and analemma set. And that's something that Paul says makes his

head hurt. So we'll stop here, with Pat cheerily putting Post-Its on her ceiling

and Paul puzzling over analemma rise. Then maybe we'll have another round of

drinks by the pool and watch a few shadows move. After all, it's science.

If you find any errors, typos or anything else worth mentioning, please send it to [email protected]

## Analemma dilemma

The equation of time, showing the contributions of earth’s orbital eccentricity and the obliquity, or tilt, of its axis. Plotted using code posted on Wikimedia Commons by Thomas Steiner.

Jupiter remains a force to be reckoned with in our early January evening skies, still high at sunset and visible until it sets, a bit after midnight.

Tuesday night, Jan 3, sees a double shadow transit a bit before midnight then the following Tuesday night, Jan 10, sees another multiple-moon event, starting around 10 p.m. with Ganymede and Europa, then following up with the same two moons’ shadows starting around 1:15 am.

Venus remains in our dusk skies. It makes a fairly close pass (a bit over a degree) with Neptune on the 13th – can you find the dim blue orb in the twilight sky so close to bright Venus? Uranus, too, is visible in early evening, though it’s much higher in the sky, in Pisces.

Mercury is visible in the dawn sky through most of the month, disappearing in the last week of January. Saturn, too, is in the morning sky, showing a ring tilt of about 14 degrees, which won’t change much until October.

Mars rises in late evening, so late-night observers can start getting their “Mars eyes” on for this year’s opposition. True “Mars season” won’t start until the opposition, on March 3, but it always takes practice to remind your eyes and brain how to see details on that small, red disk, so it doesn’t hurt to start early. Right now it’s only about 10 arcseconds – but it’ll only reach 13” at opposition, so 10” now isn’t so bad.

Last month saw the successful launch of the new, much larger, Mars rover, “Curiosity”, more formally called the Mars Science Laboratory.

Curiosity is a lot bigger than the two rovers that are on Mars now, at about the size of a Mini Cooper (the new upsized version, not the classic sixties classic) and only weighs a bit less than the car.

It’s not as fast as a Mini Cooper, though, with a top speed of around 90 m (300 feet) per hour. On the other hand, Minis don’t have radioisotope power sources that last for 14 years without a visit to a gas station, and they can’t drive themselves around another planet taking chemical analyses of rocks and soil.

Last month also saw news from one of the older rovers.

Opportunity, still rolling along just fine, detected some gypsum – considered a definite sign of water. I know, water on Mars, yawn.

We’re used to new water-on-Mars news every month or so now. But really, when you look at it, that’s one of the amazing successes of the rover program, isn’t it?

I’m writing this on a dark, cold December evening. We’re still a few weeks away from the winter solstice, the shortest day of the year.

But there’s hope – last week, December 6th, was the earliest sunset, and from now on the sun will set later, even as we move into winter.

That’s always a hard thing for me to wrap my head around – why is the date of the latest sunset happen more than two weeks before the shortest day?

It has to do with the famous “Equation of Time”. And that’s tied up with the analemma, that odd figure eight that makers of world globes like to draw in the middle of the Pacific Ocean. Usually it’s just there, without any explanation for what it means or why. And it’s too bad, because it’s pretty interesting for astronomers.

If you take an accurate clock, and every day you go out exactly when the clock says noon, take a compass and measure the sun’s position, you’d expect it to be due south, right? But you’ll find that it hardly ever is. No, you didn’t mess up and forget about daylight savings time: the problem is that the sun appears to move at different rates across our sky, sometimes faster than the clock, sometimes slower.

That’s also why sundials seldom show the right time.

The rate changes for two reasons. First is the eccentricity of our orbit. Because earth’s orbit is elliptical and not circular, earth changes speed as it orbits the sun. So depending on whether we’re in a fast- or slow-moving part of earth’s orbit, the sun will appear to race ahead or lag behind the meridian even when our clocks say it should be noon.

Second, because our axis is tilted, the changing angle of the ecliptic at different times of year adds a second complication.

The “Equation of Time” combines these two effects to compute how many minutes ahead the sun’s position (“apparent solar time”) will be compared to our smoothly-running clock (“mean solar time”). Each effect is a simple sine wave, so adding them gives us the equation you’d need to keep your sundial adjusted.

Of course, in real life there are other influences too, including the gravitational effects of other planets like Jupiter. Calculating all those effects is a lot harder, but those effects are small and normally you can ignore them, unless you’re trying to guide a spaceship or analyze Large Hadron Collider results.

So what about that figure eight in the Pacific Ocean?

The analemma has two components. Its horizontal component is the equation of time. Take that, then add a vertical component representing the sun’s declination – how far north or south is it? – and you get that familiar figure eight.

The neatest thing about the analemma is that it describes the path the sun really takes in the sky. You can photograph it! Set up a fixed camera with a wide-angle lens in your backyard, pointing south, and take a photo every day (or once a week) at exactly the same time. (Subtract an hour during daylight savings.) Combine all those frames (or make it a multiple exposure), and you’ll end up with a beautiful figure-eight analemma.

So why do they put the analemma on globes? And why in the Pacific Ocean?

Well, ship navigators who tracked their location by sighting with a sextant (in the days before GPS) needed that information, so they could compare their sightings to what their clocks said.

But it would seem easier to use a table listing values for the equation of time, and I doubt many ships were navigating using small consumer globes. In truth, I can’t figure out why they put it on globes, except for the nice old tradition linking maps and celestial navigation.

The second question is easier to answer: if you’re going to stick an analemma on your globe, put it in the Pacific because that’s the biggest place where there’s no land to get in the way.

I guess an analemma is more useful than a warning like “Here there be dragons”.

One last thing about analemmas: because they’re due to the shape of a planet’s orbit and how much its axis is tilted, not all planets have similar analemmas. Quite a few planets have figure-eight shaped analemmas like ours, but Mars’ analemma is teardrop shaped, not a figure-eight – it never crosses itself. And Saturn’s is a figure eight with a tiny top loop, so it doesn’t look like an eight at all.

So watch the sun, pay attention to those sunrise and sunset times, and think about analemmas over the next few months as you’re waiting for the warm spring weather!

## Why Earth is Closest to Sun in Dead of Winter

It's winter in the Northern Hemisphere and we're at our closest point to the Sun. Closest? Yes, you read that right. Closest. For northerners, the winter solstice has just passed. But the truth is, on January 3, 2007, Earth reaches perihelion, its closest point to the Sun in its yearly orbit around our star.

At first glance, it makes no sense. If Earth is closest to the Sun in January, shouldn't it be summer? Maybe, if you live in the Southern Hemisphere. So what does this mean?

Earth's orbit is not a perfect circle. It is elliptical, or slightly oval-shaped. This means there is one point in the orbit where Earth is closest to the Sun, and another where Earth is farthest from the Sun. The closest point occurs in early January, and the far point happens in early July (July 7, 2007). If this is the mechanism that causes seasons, it makes some sense for the Southern Hemisphere. But, as an explanation for the Northern Hemisphere, it fails miserably.

In fact, Earth's elliptical orbit has nothing to do with seasons. The reason for seasons was explained in last month's column, and it has to do with the tilt of Earth's axis. But our non-circular orbit does have an observable effect. It produces, in concert with our tilted axis, the analemma.

If you plot the noontime position of the Sun in the sky over a one-year period, it produces a figure-eight shape on the sky (Figure A). This is the analemma. You may have seen it drawn on a globe of Earth. The shape results from the combination of two things: the 23.5° tilt of Earth on its spin axis, and the elliptical shape of Earth's orbit around the Sun.

The highest point on the analemma is the Sun's noon position on the summer solstice. The lowest point marks the winter solstice. The difference in the Sun's noontime height in the sky is caused by Earth's tilted axis. What about the left-to-right variation in the analemma's curve?

That's where our elliptical orbit comes in! Look at Figure A again. Notice the vertical line running up from the south point on the horizon? That's the meridian. The meridian runs straight up and over the sky, from due north to due south.

If Earth's orbit was a perfect circle, the Sun would cross the meridian at noon every day (ignoring daylight savings time). But our orbit is slightly oval-shaped. In July, we are at our furthest point from the Sun, and Earth moves slower than average along its path. In January, we are closer to the Sun, and Earth speeds up a bit in its orbital progress. The result of this change in speed means the Sun crosses the meridian a little early, or a little late, depending on where Earth is in its orbit. For all points along the curve to the left of the meridian, the Sun is "slow." It crosses the meridian after 12:00 p.m. For all points along the curve to the right of the meridian, the sun is "fast," crossing the meridian slightly before noon.

Astronomers call this the equation of time. It is marked on many sundials. The equation of time is defined as the difference between true solar time (determined by the Sun's position in the sky) and mean solar time (the time told by your watch). The two times can vary by as much as 16 minutes over the course of a year.

Earth reaches perihelion on January 3, 2007 (Figure C). The Earth-Sun distance will be 147,093,602 km. Aphelion, the greatest distance from the Sun, occurs on July 7, 2007, when the Earth-Sun distance will be 152,097,053 km.

The difference between the two is 5,003,451 km, (3.3 percent), and not enough to cause the seasons. Even though, at this time of year, we're as close to the Sun as we can get, for the Northern Hemisphere, it will always be winter.

## Why would the Earth's solar analemma would be still a figue eight even if Earth's orbit was circular? - Astronomy

The analemma (the distorted figure-eight pattern) is more than 46 degrees long. The sun is actually only about 1/2 degree wide in the sky. It's the same size as the moon (hence eclipses), and easily covered by your index finger at arm's length. Better to test that statement with the moon! The sun seems bigger because it's so bright.

The camera never moved. It was kept on a tripod on a windowsill in a hallway of Carole Sandner Hall for the whole year. I didn't even touch it. Wires connected it to a remote shutter control and power. A USB cable allowed downloading of the sun pictures to a computer. A lead weight was hung from the tripod for stability. The setup is shown below.

It was enclosed in a foam-board box. Thanks to Fr. Tim Scully, C.S.C. for permission to set it up!

I wrote a MATLAB program which calculates the position of the sun in the sky for any longitude and latitude on a specified dates and times. (The program is in the book as an example.) The coordinates of the window were determined using GPS and the elevation angles of the dome and surrounding buildings were measured with a sighting inclinometer. Magnetic compasses proved unreliable around the building, presumably because of metal in the walls. To find the compass direction pointing from the window to the center of the dome, I took a series of photos while the sun moved over the dome one day in early Fall and noted the timestamp of the photo which showed the sun directly over the statue of Mary. Using the MATLAB program and that date and time, the compass angle from window to dome could be determined.

The MATLAB code was then used to generate a plot of what the sun's position would be throughout the year at specific times of day, and a simple geometrical model of the dome. I chose a time such that the sun's path would wrap the dome, but stay above the roofline at the lowest point in the winter. The plot of predicted positions that led to the choice of time is shown below. The analemma bending is just due to plotting "flat" as a function of azimuthal and vertical angles.

No, the remote shutter control has a timer built into it, so I just set it to take an exposure every 24 hours. Because of drift in the timer, it needed to be reset once a week. My normal intervention was just to reset the timer to USNO time each Saturday morning.

When the sun was visible, an image taken though the solar filter was just a round solar disk on a black background. Some days were cloudy and there was no image. It's South Bend, Indiana, after all. I wrote a MATLAB GUI Tool that let me select roughly evenly-spaced images from among the sunny day photos. The program then combined them by simply taking the maximum value of each pixel. The composite sun image was then combined with the original foreground image, taken without the solar filter but from the same camera position, and tweaked, using PhotoShop (levels and curves).

You see the same analemma shape no matter where you are on the earth. The ND Analemma photo is taken in the morning so the figure is tilted to the east. Near local noon it would be upright, as in this photo. In the evening it tilts west.

If the earth's orbit were circular the analemma would be a symmetric figure-eight. The explanation, a somewhat complicated bit of geometry, can be found here. The distortion of the symmetrical figure-eight is because of the elliptical nature of the earth's orbit around the sun.

### Planet - What is the fifth moon-like object I saw around jupiter through my telescope?

Tonight while observing Jupiter I was able to see a five (what looked like) moons. I have observed Jupiter often before and only been able to make out the four galilean moons. It's my understanding that through my 8-inch scope it would be impossible for me to see any others. Switching to a widefield lens I was able to see Jupiter along with some nearby stars however the 5 moon-like points around it all still appeared. well, moon-like in comparison to the bright stars.

I observed Jupiter from about 2016-04-15 05:20 to 05:40 UTC and all five of the moon-like objects were present the whole time. What was I probably looking at?

## Solar analemma 2015

A compilation of images of the Sun taken at the same time and place over the course of 2015, as seen from Sulmona, Abruzzo, Italy. Credit: Giuseppe Petricca

If you took a picture of the Sun every day, always at the same hour and from the same location, would the Sun appear in the same spot in the sky? A very fine image, compiled by astrophotographer Giuseppe Petricca from Italy, proves the answer is no.

"A combination of the Earth's 23.5 degree tilt and its slightly elliptical orbit combine to generate this figure "8" pattern of where the Sun would appear at the same time throughout the year," said Petricca.

The analemma is considered by many to be one of the most difficult and demanding astronomical phenomenon to image. Astrophotographers need to dedicate an entire year to the project. It requires diligence to take images 30 to 50 times throughout the year at the same time of day and same location.

It is interesting to note that analemmas viewed from different Earth latitudes have slightly different shapes, as well as analemmas created at different times of the day. Also, analemmas on the other planets have different shapes.

If the Earth were not tilted, and if its orbit around the Sun were perfectly circular, then the Sun would appear in the same place in the sky throughout the year. But then, we also wouldn't have seasonal change, so I vote to keep axial tilt!

Petricca combined 32 pictures of the Sun taken at 12pm local time throughout the months and seasons, all shot with the same settings and exposure times (ISO 100, f/8.0 and 1/1000″ exposure time).

A compilation of images of the Sun taken at the same time and place over the course of 2015, as seen from Sulmona, Abruzzo, Italy. Credit: Giuseppe Petricca

"I was lucky to have the last year with good sunny skies at the right times, even if some months were really difficult to image," he explained via email. "The background view is the one from the first picture, January 4th, 2015, after three days of snow."

Petricca used a Nikon Coolpix P90 Bridge Camera mounted on a fixed tripod, with images taken from a field nearby his home Sulmona, Abruzzo, Central Italy. "To take pictures of the solar disk I used an Astrosolar filter in front of the camera, then I composed the analemma digitally, via Photoshop CC," he said.

• The Opportunity rover captured this analemma showing the Sun’s movements over one Martian year. Images taken every third sol (Martian day) between July, 16, 2006 and June 2, 2008. Credit: NASA/JPL/Cornell/ASU/TAMU
• The analemma created by Giuseppe Petricca, annotated with the dates each picture of the Sun was taken. Each image of the Sun was taken at the same time and place over the course of 2015, as seen from Sulmona, Abruzzo, Italy. Credit: Giuseppe Petricca