Secrets of the Swallows

Surviving for millions of years while other species became extinct, according to one theory advanced in the '60s, the swallows may have learned to rely upon a natural solar-induced phenomenon for their continued existence. The daily shadow tip path of the Sun, as cast by any given Earth prominence, forms a family of conic curves during the course of the year tracing hyperbolas, parabolas, ellipses and straight lines. When the curves become a certain shape the swallows (according to the theory advanced) depart their winter home near Buenos Aires and begin their trek to San Juan Capistrano arriving at the time of spring. In reality, the cue that many animals rely upon on when to migrate is governed by a circadian mechanism (internal biological clock). The swallows begin their trek back as the lengths of the days and nights begin to become equal. This corresponds to a unique characteristic of the shadow tip path of the Sun at this time. What shape is the shadow tip path becoming at the swallow's departure from Buenos Aires.

a. concave up
b. concave down
c. straight line
d. elliptical

The answer is C.

First you must know that the swallows return to San Juan Capistrano yearly at spring. The shadow tip path cast by any prominence generates a family of conic curves during the course of the year and is concave up during the winter degenerating to a straight line on the vernal and autumnal equinox and becoming concave down during the summer (the reverse is true in the southern hemisphere). At spring or the fall, the plane containing the axis of the Earth is perpendicular to its orbital plane and its radius to the Sun. At this time there is no tilt toward the Sun and the shadow tip path curve has degenerated to a straight line. The rest of the year there is a varying tilt component toward the Sun that results in a unique conic curve (hyperbolas, parabolas and ellipses) as seen in Figure 1. Figure 1 illustrates the conic curve that is generated depending upon the relationship of the declination of the Sun and the latitude of the observer or the declination of the Sun and the inclination of the receiving surface to the horizontal plane.

The shadow tip path geometry is a function of the declination of the Sun and the latitude of the observer. Figure 2 depicts a cross section of the Earth with the Sun S on the observer's meridian. The observer's zenith is at Z and his latitude L is arc QZ. If the polar distance of the Sun (90°-d) is equal to the observer's latitude, the shadow tip path will be a parabola; at higher latitudes it will be an ellipse and lower, a hyperbola. In any event these curves flatten to a straight line as the equinox approaches. The case shown illustrates the Sun at the summer solstice with declination of 23.5° and the latitude of the observer 67.5° (when the latitude equals the Sun's polar distance)

Figure 2. Shadow tip path geometry guide

The shadow tip path cast by the Sun resulting in a hyperbola is shown in Figure 3.

Figure 3.

The locus of the shadow tip path of the Sun was recognized as the basis of a sun dial and calendar early in history. Additionally the properties of the shadow tip path can be used as the basis for a solar compass as well which is depicted in Figure 4. The solar compass depicted allows the observer to determine true North and the local apparent time. The solar compass is held level and rotated until the tip of the shadow, cast by the raised index, is touching the day of the year curve shown on the face of the compass enabling the observer to read the direction of true North and the local apparent time.

Figure 4. Solar Compass for 35° to 45° north latitude

30° - 40°

Bibliography

Dorrie, Heinrich. 100 Great Problems of Elementary Mathematics Their History and Solution. New York: Dover Publications, Inc.1965

Rohr, Rene. J. Sundials, History, Theory and Practice. Toronto: University of Toronto Press, 1965