Posts Tagged ‘analemma’

03/13/2017 – Ephemeris – More thoughts about yesterday’s time change

March 13, 2017 Comments off

Ephemeris for Monday, March 13th.  The Sun will rise at 7:58.  It’ll be up for 11 hours and 48 minutes, setting at 7:46.  The Moon, 1 day past full, will rise at 9:03 this evening.

We are now plunged back into dark mornings like we were two month’s ago thanks to the start of Daylight Saving Time.  However we are only a week from the vernal equinox, the first day of spring here in the northern hemisphere.  However some of my blog followers down under will experience the start of autumn on that day.  For us in the next three months the sunrise time will back down 2 hours, and will rise around 6 a.m.  Our sunset times will advance a bit less than that, an hour and 45 minutes.  The lopsidedness is a consequence of both the Earth’s axial tilt and its slightly elliptical orbit.  We are moving somewhat away from the Sun now and are slowing down a bit.  It’s all kind of hard to explain, but makes perfect sense… eventually.

Times are for the Traverse City/Interlochen area of Michigan. They may be different for your location.


Well, I’m going to try to explain it now.

If one visits most observatories, there will be a clock, usually in the dome that doesn’t seem to read the correct time. We have one in Northwestern Michigan College’s Rogers Observatory’s dome. It only agrees with the time on your watch for an instant on October 16th. It’s called a sidereal clock and it measures Earth’s rotation with respect to the stars, and it gains approximately 4 minutes a day compared to our normal clocks which are geared to the Sun.

The Earth and all the planets orbit the Sun in a counterclockwise direction when seen from the north. Also the Earth and most of the planets spin also in a counterclockwise direction. The exceptions are Venus and Uranus. Most large satellites like our Moon also orbit their planets in a counterclockwise direction. From the surface of the Earth these bodies appear to generally move eastward, while the celestial sphere mirror reflects our eastward rotation by appearing to move east to west.

The Sun appears to move about one degree a day eastward. (360° / 365 days = 1° approximately). The Earth rotates through 15° an hour (360°/24 hours = 15° and 60 minutes in an hour / 15 = 4 minutes to rotate a degree). So on average and approximately after the Earth rotates back to the same point with regard to the stars, it has to rotate and extra degree to catch up to the Sun, which takes 4 more minutes, which is why the solar day is 4 minutes longer than a sidereal day. (Well, 3 minutes 55.9084 seconds if you want to get picky about it!)

Of course it’s not that simple. It never is that simple. This would all work out if the Earth orbited the Sun in a uniform circle and the Earth had no axial tilt. The Earth’s tilt is also called obliquity. The Sun would appear to move uniformly over the Earth’s equator. That Sun, called the mean Sun is what we base our solar time on, not the real Sun. However the Earth’s orbit is elliptical, with the Earth moving fastest at its perihelion or closest point to the Sun, around January 3rd, and slowest at aphelion or farthest point around July 4th. Also the Earth’s axial tilt is 23 ½ °, and is only on the equator two days a year March 20th and September 23rd.

Ever see this funny figure 8 in the Pacific Ocean on old globes?

The Analemma

Illustration 1. An Analemma graphically demonstrating the actual Sun’s relation to the mean Sun during the year.

Both eccentricity and obliquity work together to produce the analemma as seen in the diagram below. This figure 8 can actually be photographed in the sky by exposing the same frame of film at regular intervals, or stacking images of the same area of the sky over a year at the same time of day. There are plenty of examples using an Internet search engine to search for analemma images. It is one way to illustrate the equation of time, which is the correction one must make to a sundial reading to get to the correct local mean solar time. To that one must add or subtract one’s offset from the time zone’s time meridian. See Friday’s post.

Cause of the analemma

Illustration 2. How eccentricity of the Earth’s orbit and obliquity combine to affect the analemma. The effects add at the bottom near the winter solstice and subtract near the summer solstice.

The above diagram was taken from Ethan Siegel’s Starts with a Bang blog: which also explains it.

The equation of time can be found in tabular form for easy sundial correction, or in a linear graphical form as seen below.

Equation of Time

Illustration 3. Linear representation of the equation of time

Actually the biggest effect on the equation of time, is especially near the solstices is the Earth’s obliquity (axial tilt). Below we see how the Sun’s declination affects how fast it appears time wise.

Sun crossing time lines

Illustration 4. How the Sun’s declination affects how rapidly it appears to cross time lines (meridians).

Declination of celestial objects is the same as latitude on the Earth. A star whose declination is the same value as one’s latitude will cross at the zenith once a day. In illustration 4 note that the near the solstices the time lines (meridians) are closer together, so the Sun will pass them faster than when near the equinoxes where the time lines are farther apart and the Sun is moving somewhat diagonally, taking longer to cross the time lines. In time only east-west motion counts.

Looking at Illustration 2, I’ve added arrows at the top (northern) and bottom (southern) extremities of each analemma source to give some idea of the Sun’s apparent speed at the solstices. At the bottom, near the December solstice the eccentricity speed of the Sun adds to the obliquity speed increasing the effects at that part of the analemma lobe, making it bigger. At the June solstice end of things eccentricity speed is in the opposite direction, slowing the Sun down.

So what? This affects the dates of the earliest and latest sunrises and sunsets. Here are those dates and time values for us here in northern Michigan (specifically the Interlochen/Traverse City area):

Earliest and Latest Sunrises and Sunsets

Table of Earliest and Latest Sunrises and Sunsets during the year for Interlochen/Traverse City area of Michigan.

All this may make little difference to our modern lives, governed by the atomic clocks in Paris and Fort Collins, Colorado, divorced as they are from the Earth’s actual rotation and the Sun except for the inclusion of the occasional leap second, like we had last December 31st. To folks like me who are amateur astronomers and have (or had in my case)  a day job, it would’ve been nice to have, on the summer solstice, astronomical evening twilight end before midnight.

I hope this helped rather than confused you.  What do you think?  drop me a comment.

More questions about the length of daylight hours

January 12, 2015 1 comment

This is the result of a question I got about why the daylight hours change the way they do during the year.  My answer is posted here as “How come hours of daylight changes very slowly around the solstice, but very rapidly around the equinoxes?”

My correspondent has a few more questions.  I’ll boil them down.

I pretty much understand why daylight changes rapidly at the equinoxes and slowly at the solstices based upon your map showing the ecliptic and how the steepest part is at the equinoxes. Also, the figure eight drawing makes sense. But why does the curve of the ecliptic seem to linger for a time at the solstices before plunging? Does it have to do with the speed of the Earth in its orbit?

The analemma, as seen below, is the result of two phenomena.  First, the tilt of the Earth’s axis which would on itself make a figure 8 with equally sized lobes, with crossing point at the equinoxes.  Second, the Earth’s orbit of the Sun is a slight ellipse, meaning for our purposes here that the Earth moves its fastest near perihelion when the Earth is nearest the Sun, around January 4th. and slowest at aphelion, when the Earth is farthest from the Sun, around July 4th.  That makes the bottom lobe larger because the Sun is by reflection moving faster eastward in the sky.  The apparent slowness that the questioner perceives is an illusion because the Sun appears to be moving in a more directly eastward, and changed the actual time of local solar noon.  Wikipedia has a detailed discussion of the analemma.


This figure 8 is called an analemma. One can find it on old globes in the Pacific Ocean. Explanation below. Created using my LookingUp program.

I had stated in the prior post that daylight hours would be 12 hours at the equinoxes and also all the time at the equator.  So here’s the other question.

At the equator, day length does change over the course of the year, doesn’t it? At the equinoxes it would be 12 hours long, but at the summer solstice up north it would sink towards the south by 23 degrees and at the summer solstice in the south it would sink towards the north by the same amount.

Other than getting cooperation from someone who either lives on or has visited the equator, I generated a calendar of sunrise and sunset times for the equator, specifically for 0º longitude and 0º latitude.  A link to it is here.  Also read the explanation on that calendar page.

The answer is No, the daylight hours at the equator doesn’t change over the year.  The one minute variance has to do with the Analemma.

12/02/2014 – Ephemeris – The unequal dates of latest sunrise and earliest sunset

December 2, 2014 Comments off

Ephemeris for Tuesday, December 2nd.  The sun will rise at 8:00.  It’ll be up for 9 hours and 3 minutes, setting at 5:03.   The moon, 3 days past first quarter, will set at 4:41 tomorrow morning.

This evening’s sunset is just a minute from the earliest sunset of the year.  The Earliest sunset will actually be on the 9th.  However the latest sunrise won’t occur until January 2nd.  The reason combines the effects of the tilt of the earth’s axis and the fact that the Earth is only a month from perihelion, its closest to the Sun.  Both these effects cause the sun to appear to move faster eastward than average, so the Earth has to rotate a bit farther each day to catch up with the Sun.  This makes the sunrise and setting events later than one would expect, so they don’t occur together on the shortest day of the year, the 21st this year.  Our sunrise this morning is still 19 minutes earlier than the latest sunrise on January 2nd, 2015.

Times are for the Traverse City/Interlochen area of Michigan.  They may be different for your location.



This figure 8 is called an analemma. One can find it on old globes in the Pacific Ocean. Explanation below. Created using my LookingUp program for Traverse City, MI near +45° latitude.

The analemma is a graphical representation of a daily value called the Equation of Time.  It’s best known use is in corrections to sundial time.  The vertical axis is the sun’s declination or north-south position.  It is highest at summer solstice and lowest at winter solstice.  It is the result of two effects:  the tilt of the Earth’s axis to the plane of the Earth’s orbit around the Sun, and the change in the Earth’s velocity around the Sun as the Earth moves from perihelion, its closest to the Sun in early January to aphelion, its most distant in July.

If the Earth’s orbit were circular, and it orbited the Sun at the same speed.  The analemma would be skinner and the north and south lobes would be of equal size.  Since we’re closer to the Sun in the winter, we move faster than average around the Sun, so it appears to move faster eastward.  That combines with the faster appearing movement of the sun crossing the closer hour lines at higher and lower declinations.  In the diagram above note that the vertical hour lines are slightly closer together at the bottom and the top, so the Sun, moving eastward each day crosses them quicker.  Near the winter solstice the two effects work together making sunrise and sunset trending to be later than normal.  For the summer solstice the eastward speed of the sun is slower than normal, because we’re farther from the Sun.  This works against the effect of the earth’s tilt but cannot completely negate it, making the top of the loop smaller than the one at the bottom.  The arrows show the speed and direction of the Sun at the solstices.

To see real analemmas search for analemma images on the Internet.  It takes a year to photograph one.

Categories: Concepts, Ephemeris Program Tags: