Mon 2020-Dec-21

Analemma, my Analemma, by Douglas W. Dodds

Tagged: Math TheDivineMadness

This is one of the annual emails the Symbolics lisp machine community used to get from Doug Dodds:

Analemma, my Analemma
Douglas Dodds ( or
Cambridge, Mass., USA

When I came to Boston from St. Louis, I first had to adjust to the trauma of the local climate. A bit later, another environmental difference became evident: the hours of daylight and darkness. Two memories of my first year at MIT epitomize the difference. I remember seeing the red sun five minutes from setting at 4 pm on an early December afternoon; I realizied that it sure had been getting dark early. And I remember staying up most of the night studying in late April, and being astonished that daylight was breaking in the east at 3:45 am!I soon understood that Boston’s more northerly latitude resulted in a larger excursion in length of daylight than I was used to (roughly between 9 hours and 15 hours); and that its position relatively far east in the Eastern Time Zone shifted the whole day toward the morning on the clock (local mean solar noon here occurs 16 minutes before noon, EST). I love the long, light summer evenings here, but have always been depressed by the early darkness in winter.In recent years, I got a bit interested in astronomy, and discovered a subtler effect: the actual solar time cyclically speeds up and slows down relative to mean solar time over the course of the year! During most of the year the deviation of actual from mean solar time is small and slow, but between November and February, sun time travels from its maximum “fastness” to its maximum “slowness”, a total excursion of 30 minutes! One result of the wintertime swing in relative solar time is that although, as everyone knows, December 21 (approximately) is the shortest day of the year, the day of the latest sunrise is almost two weeks later, on January 4. And, most important (fanfare), the earliest sunset occurs on December 8! Amazingly, the sunset actually begins, ever so slowly, to become later after that date.I am seldom concerned with when the sun rises, late as that is during the Boston winter; the time of sunset defines my length of daylight. So since that discovery, it has always been a cheering consolation to me that the “day” is already at its shortest on December 8, before winter really has set in! For me, the light returns already; it makes winter a little easier to face.

Why the Daylight Period Varies

The strange and fascinating wandering of actual solar time relative to the clock is expressed in a peculiar parametric curve called the analemma. It shows (now listen carefully) the locus of the subsolar point on the Earth’s surface at a given Universal Time, for all days of the year. Alternatively, it is the locus of the sun in the sky at a given clock time, say 9 AM, on all days of the year.If one were to strobe the sun at the same clock time every day for a year, the sun would trace out its analemma in the sky. Somebody has actually done this! In a photograph that is now a classic, Dennis di Cicco of Sky and Telescope magazine photographed the sun on the same plate (through a filter), with a permanently mounted camera, about every ten days, at exactly the same time of the morning. He then removed the filter to shoot the background in normal daylight; the result was a cluster of suns rising over his neighbor’s house, in the bottom-heavy figure-8 shape of an analemmaic curve.The analemma is a parametric curve, plotting north-south latitude on the Y-axis, deviation in east-west longitude on the X-axis, and the parameter of graphing is the calendar date. An alternative interpretation of the X variable is time, via the Earth’s rotation speed of one degree of longitude every four minutes. If there were no deviation of solar time, the analemma would just be a vertical line, tracking the sun’s seasonal movement from about 23 degrees north of the equator to 23 degrees south.The actual looping analemma is due to the sum of two effects.

  • The first has to do with the seasonal apparent travel of the sun north and south. The sun (or rather the subsolar point) travels at a constant speed along a wavy (sinusoidal) path. It is roughly at a northern plateau around the June solstice, moving almost parallel to the equator. It then begins to angle south, crossing at the September equinox, then approaching a southerly plateau in December, and so on. The speed along the path is constant, but the longitudinal (east- west) travel of the sun against the coordinates of the sky is the projection of this speed on the equator. Clearly, this cycles faster and slower. Around the solstices, the projected motion is at its fastest, around the equinoxes at its slowest. The result of this effect alone would be a propeller-shaped analemma, a skinny, equal-looped figure-8.
  • The other effect, completely independent, is due to the fact that the earth’s orbit is not a perfect circle, but is slightly elliptical. The Earth moves a little faster when travelling the portion of its orbit that is closer to the sun, a little slower on the more distant part. So the advance of the sun across the firmament varies correspondingly faster and slower through the year.

By coincidence, the ellipticality effect is, at this epoch, almost synchronized with the seasonal motion of the sun: perihelion (Earth closest to sun) is on January 3, the southern solstice on December 21. So the resulting curve from the elliptical orbit alone is a thin elliptical analemma tilted slightly to the northwest-southeast.The complete analemma is the sum of these two curves. The longitudinal extents of the two are roughly equal; and they are in phase (reinforcing) on the southern half (our winter) and at opposite phase (cancelling) in the northern. The result is a distorted figure-8 curve with a very fat bottom loop, a small top loop, and a slight scrunch to the right.The time deviation of the sun is slight from April to September, less than five minutes fast or slow. But the sun reaches its maximum fastness (16 minutes) in mid-November (sunrises and sunsets are earlier than average). And it reaches its maximum slowness (14 minutes) in late February (sunsets later). The period around the December solstice is a headlong rush of the daylight toward the evening. The result is the widely spread latest-sunrise and earliest-sunset times that I enjoy so much, while the northern winter sets in.

My main reference for the information in this piece is an excellent article on the analemma by Bernard Oliver, in Sky and Telescope, July 1972.

— Doug Dodds

Published Mon 2020-Dec-21