Observing Project 6C The Analemma and the Equation of Time

Have you ever looked at a globe of Earth and noticed the odd figure eight shape that is placed usually somewhere over the Pacific Ocean and wondered what it was? The figure eight pattern is called the analemma. The analemma is a representation of where the Sun appears in the sky at exactly the same day on each day of the year. Two important motions that are fundamental to astronomy and to our lives on Earth are responsible for the motion. These motions are Earth's axial tilt and Earth's elliptical orbit around the Sun.

Earth's axial tilt causes the Sun to alternately travel north, then south in the sky reaching extremes of 23.5 degrees north of the equator in June and the same distance south of the equator in December. So if the Sun's position on the sky were to be plotted each day starting in December, it would move a bit farther north each day until June 21, then it would stop and begin moving south again. So this explains the up/down pattern on your globe. Now why the left/right motion?

If Earth moved in a perfectly circular orbit around the Sun, then it's orbital speed would be constant at all times. Then each day at local noon, the Sun would cross the local meridian. But Earth's orbit around the sun is slightly elliptical, so while Earth's rotational speed is constant its orbital speed is not. In January when Earth is at perihelion, it is traveling at its fastest around the sun. After 24 hours, Earth rotates through 361 degrees completing one solar day.18 But because Earth is moving slightly faster than normal it takes slightly more than 24 hours for the Sun to catch up. Now it will only take about eight seconds for Earth to rotate the additional distance necessary to bring the Sun to the meridian. What is important to understand is that difference is cumulative and builds up each day until Earth's orbital speed slows down enough so that the actual solar day is an even 24 hours again. During that time the Sun is drifting slowly eastward in the sky each day at local noon. By early spring the cumulative delay in the Sun's crossing the meridian will have built up to approximately eight minutes. The difference between local noon and the time when the Sun crosses the meridian is called the equation of time. After April 2, the effect is reversed because Earth is slowing down in orbit as

18 A solar day is the time from noon to noon and is exactly 24 hours. The time it takes Earth to complete one full rotation (360 degrees) is slightly less than that, 23 hours and 56 minutes. The difference is caused by the additional one degree of rotation needed to make up the change in point of view caused by Earth's motion around the Sun.

it drifts farther from the Sun. This will cause the Sun to drift westward in the sky until October. This turns the straight north-south line of the Sun's seasonal motion into the figure eight illustrated on the globe.

One of the most difficult photographic achievements in astronomy is a photograph of the full analemma. This will require the use of a camera mounted on a stationary pier that will not move over the course of the entire year. The camera must be able to open the shutter repeatedly without having to advance the film. Probably the right camera for this task would be an older model Hasselblad. Such cameras have not been built for many years, so just the difficulty in finding the right equipment makes imaging the analemma very difficult. The next thing you then need is a lot of luck. You want to be able to image the Sun at regular intervals, approximately once each five days. The right time to begin your project would be at the equinox, when the Sun is at the crossing point of the figure eight and fairly early in the morning. This provides smooth air and maximizes the probability of clear skies. What you must account for next is that the two loops are not the same size on film. When the Sun is in the south, the low altitude will cause the lower loop of the analemma to appear much larger than the upper one so you must position your camera so that the upper loop is targeted into the upper right of the frame. If you are successful, when you are done you will have achieved something that few astronomers at any level have ever accomplished, an image of the Sun's annual journey across the sky.

There is a simpler way of documenting the analemma if you don't want to depend on luck and spend a lot of money on photographic equipment. Find a spot where the Sun shines all year around without ever being hidden behind a house, a tree or anything else where you can drive a rod in the ground. The rod should be about three feet tall and be constructed in such a way that it will not move over the course of the year. Each few days at exactly the same time (noon at local standard time works best and one hour later during daylight savings time) note where the end of the shadow cast by the rod falls and mark that spot. Do this for a year and when the year is over, your marks on the ground will have traced the analemma.

The analemma also tells us something important about our Earth. The figure eight shape is very narrow and the length of it only spans some 47 degrees of sky. This tells us that Earth's orbit is very close to circular and the change of seasons is not unusually extreme. Imagine what the analemma might look like if Earth's orbit were as eccentric as Pluto's were and its axis were tilted to the extremes that Uranus' was. The extremes of season and temperature between perihelion and aphelion would make life on this planet very difficult to enjoy at best.

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