Calculating The Equation of Time
As noted in the introduction to this article, for many purposes the equation of time is obtained by looking it up in a published table of values or on a graph. For dates in the past, such tables are produced either from measurements done at the time, or by calculations. Of course for dates in the future, tables can only be prepared from calculations. Also, in devices such as computer-controlled heliostats, the computer is often programmed to calculate the equation of time whenever it is needed, instead of looking it up. Two types of calculation are currently in use, numerical and analytical. The former are based on numerical integration of the differential equations of motion that include all significant gravitational interactions and relativistic effects. The results are accurate to better than 1 second of time and form the basis for modern almanac data. The later are based on the known solution of the equations of motion that include only the gravitational interaction between the Sun and Earth. These produce results that are not as accurate as the former, but they can be understood more easily and the equations illustrate the functional dependence on the governing parameters. Their accuracy can be improved by including small corrections, but in the process concurrent increases in complexity are incurred.
The following discussion describes a simple yet reasonably accurate (agreeing with Almanac data to within 3 seconds of time over a wide range of years) algorithm for calculating the equation of time that is well known to astronomers. It also shows how to obtain a simple approximate formula (accurate to within 1 minute of time over a large time interval), that can be easily evaluated with a calculator and provides the simple explanation of the phenomonon that was utilized previously in this article.
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