# Big figure eight in the sky

Determining when to shoot local apparent noon (LAN) requires a knowledge of something called meridian passage. This is the time when the sun will pass over a given meridian. Navigators usually find this time by examining the Nautical Almanac’s daily pages.

There is, however, an alternative method for determining the time of LAN, as well as the sun’s Greenwich Hour Angle (GHA) and declination: a simple diagram, shaped like a lopsided figure eight, called an analemma. Reflecting their importance in determining time, analemmas are often found on maps and globes.

Before the advent of rapid transportation and communication, noon each day was accepted as being halfway between sunrise and sunset. The length of each day varied with the seasons, but that didn’t significantly disrupt activities since timing to exact minutes and seconds was rarely necessary.

However, trains need to run on tight schedules; so, in the late 1800s, the railroad companies were the prime movers behind time standardization. A significant hurdle in implementing a system of accurate and standardized time lay in deciding how to describe the sun’s daily and yearly apparent motion.

The sun’s movement actually results from the Earth’s motion. There are two major components to that motion: The Earth hurtling along in its elliptical orbit, and its daily rotation on its axiswhich is tilted 23.5° to the ecliptic. These two factors cause the apparent sun to vary its motion across the sky. A few times a year the apparent sun crosses the Greenwich meridian at noon, but it usually crosses either a few minutes before or a few minutes after noon.

Scientists have compensated for these irregularities in the sun’s motion by inventing an imaginary sun that moves across the sky with precision. This sun, dubbed the mean sun, is used as the basis for solar time. The mean sun’s path across the sky is along the celestial equator, and it travels at a constant 15° per hour (360° divided by 24 hours = 15° per hour).

On a daily basis, the difference between time based on the mean sun and time based on the apparent sun is called the equation of time. This value can be found in the Nautical Almanac daily pages.

The equation of time can also be found in graphic form on the analemma. An analemma shows the equation of time on the horizontal axis and the sun’s declination on the vertical axis. The largest values for the equation of time occur around the beginning of November (16 minutes) and beginning of February (14 minutes).

The equation of time itself results from two factors: 1) the eccentricity of the Earth’s orbit and 2) obliquity of the sun’s apparent motion as measured on the Earth’s surface.

The eccentricity of the Earth’s orbit causes the Earth to change its orbital speed over the course of the year. This varying speed is described by Kepler’s second law, which states: “A radius vector within an ellipse sweeps through equal areas in equal time intervals.” Thus, the Earth’s speed at perihelion (Earth’s closest point to the sun) is greater than at aphelion (farthest distance from the sun). This causes the apparent sun to speed up and slow over the course of the year.

Maximum separation of the mean and apparent suns occurs near the beginning of April and beginning of October. At these times, the difference is approximately eight minutes.

The obliquity component of the equation of time is measured by comparing the mean sun moving along the celestial horizon with the actual sun moving along it’s true path: the ecliptic. Since we measure the sun along one axis, but it actually moves along another one, there is a certain amount of parallax error. The difference in these two motions reaches a maximum value of ten minutes around the beginning of February, May, August, and November.

Both the eccentricity and obliquity components can be plotted and then combined, producing a curve that represents the equation of time. When we examine the equation of time curve for the beginning of February, say, we see 14 minutes of equation of time. This results from adding eight minutes of obliquity factor with six minutes derived from eccentricity.

The analemma is pear-shaped and not a symmetrical figure eight because of the Earth’s elliptical orbit. The sun’s apparent motion is faster in the winter months (northern hemisphere), so there is a greater difference between mean and apparent time during this period.

When viewing the analemma, time differences to the left of zero minutes of equation of time indicate that apparent time is ahead of mean time. Thus “noon” for the actual sun will occur before mean solar noon. Times to the right of zero minutes of equation of time indicate that apparent time is behind mean solar time.

How does an analemma help navigators? It allows one to easily calculate local apparent noon, and to determine the sun’s declination and even a rough GHA. The analemma gives us the equation of time for the appropriate day. We can apply that time to “noon,” either adding or subtracting the amount, and after accounting for longitude, the time of LAN will be determined.

An analemma is both a visual aid to understanding the sun’s apparent motion with respect to the Earth and a source for the equation of time, declination, and even rough GHA. With analemma and sextant, a navigator in a raft or on an island could easily determine his latitude. Michael Carr