SPACING OF MARKS AT THE SOLSTICE

At the exact solstice, all observation marks will completely overlap since the changes in declination and elevation of the sun have essentially stopped.

At sunrise and sunset the movement of the sun (at the latitude of San Francisco) for one day before and one day after will be by an angle of approximately 1/2 arc minute each day. This is approximately 1/60 of the diameter of the sun. Thus, to observe this very small amount of motion requires that you be able to detect a sliver of light 1/60th of the diameter of the sun. This is just barely possible by shadowing out the remainder of the image of the sun, and was apparently the method of choice by many of the ancient peoples.

The daily change in elevation is somewhat approximately the same.

While it may be possible to observe such small changes, the usefulness of any system where the observations result in a line marking on a marking surface will be limited to the thinness of the lines, and their spacing so as to not overwrite each other.

SPACING CALCULATION

If the distance from your shadowing object to your marking surface is 10 feet, then the spacing for marks made 0.5 arc minutes apart is calculated as follows:

From trigonometry: The sine of an angle = length of the opposite side divided by the length of the hypotenuse, where the opposite side is the line spacing and the hypotenuse is 10 feet, or 120 inches.

sine 0.5 arc minute = approximately 0.5 x sine 1 minute = 0.5 x 0.00029 = 0.000145

Line spacing = 0.000145 x 120 inches = 0.0174 inches = about 1/50th inch.

Clearly, it would be very difficult to make and then differentiate between marks made with only this amount of spacing.

NOTE: The above calculations assume that your marking board is perpendicular to the sun's rays. If not, then the marks will be spread out, the amount depending upon the how far the board is tilted from perpendicular. However, the shadow will also be less bright due to this spreading.

MARKINGS FOR MEASUREMENTS OF THE EQUINOX

While the elevation and declination of the sun stands still at the solstices, it moves the fastest at the equinoxes, with a movement of approximately 0. 4 degrees (24 minutes) per day.

For this case the line spacing for a 10 foot shadow would be:

sin 24 minutes = 0.00698

Line spacing = 0.00698 x 120 inches = 0.84 inches

Thus, measurement of shadow height to determine the exact date for the equinoxes is feasible without using precision methods.