Unit 5: Exploring the Nature of Astronomical Phenomena in the Context of the Sun
Unit 5: Exploring the Nature of Astronomical Phenomena in the Context of the Sun/Earth/Moon System
IX. Estimating the Tilt of the Earth
How curious are you about the tilt of the Earth? This section provides a detailed account of geometric arguments that underlie a surprisingly simple way to estimate the tilt of the Earth’s axis. The only equipment you need would be a post and a way to measure its height and the length of its shortest shadow during equinoxes and solstices. If you enjoy puzzling through such geometrical arguments, read on. If not, perhaps skim this briefly to see what is here and then go on to the next section.
A. Developing and using mathematical representations to estimate the tilt of the Earth’s axis
Ancient peoples noticed seasonal changes in the lengths of shadows as well as in the Sun’s apparent path across the sky, particularly during equinoxes and solstices. A Greek astronomer, Eratosthenes (276-195 BC), for example, estimated the magnitude of the tilt by using such observations and his knowledge of geometry.
Question 5.38 How can one estimate the tilt of the Earth’s axis of rotation?
This section develops a geometrical approach to estimating the tilt of the Earth’s axis of rotation. This approach is surprisingly simple:
- Measure the maximum angular altitude of the Sun, angle α (alpha), at a given location during a solstice and during an equinox; then subtract one angle from the other.
- Or measure the maximum angular altitude of the Sun, angle α (alpha), at a given location during both solstices; then subtract the angle during the winter solstice from the angle during the summer solstice and divide by two.
- Or substitute (90º – the latitude of the gnomon) for the maximum angular altitude of the sun at an equinox and use this along with the maximum altitude of the Sun at one of the solstices as indicated in the box below.
Estimating those angles involves simply using a protractor, consulting data sources such as the tables presented in Figs. 5.56-5.59, or measuring the height of the gnomon (a post) and the length of its shadow at solar noon, dividing one by the other to find the angle’s tangent, and identifying the angle with a calculator or trigonometry table.
1. Envisioning the tilt of the Earth’s axis of rotation
The tilt of the Earth’s axis is known as its obliquity and is represented by angle ε (the Greek letter epsilon). As shown in Fig. 5.70, the tilt is the angle ε between the Earth’s axis of rotation, which is represented by the large dashed line from the North Pole to the South Pole, and the vertical axis, which is represented by the small dashed vertical line. The vertical axis is perpendicular to the plane of the Earth’s orbit around the Sun. This discussion assumes the Earth is spherical. The small dashed horizontal line represents the projection of the orbital plane on this cross section of a spherical Earth.
2. Estimating the tilt of the Earth’s axis of rotation
How can you find out what the maximum angular