physics4me

… and a piece of plywood

Schematic view of the elementary sundial

Thierry Lahaye
I describe how to obtain a rather good experimental determination of the eccentricity of the Earth orbit, as well as the obliquity of the Earth rotation axis, by measuring, over the course of a year, the elevation of the Sun as a function of time during a day. With a very simple “instrument” consisting of an elementary sundial, first-year students can carry out an appealing measurement programme, learn important concepts in experimental physics, see concrete applications of kinematics and changes of reference frames, and benefit from a hands-on introduction to astronomy.

1. Introduction
One of the cornerstones of introductory courses in classical mechanics is the derivation of Kepler’s laws. In particular, the derivation of Kepler’s first law, stating that the trajectory of a planet is an ellipse with the Sun located at one of the foci, is an important application of Newton’s laws to a multidimensional problem. However, very few students are aware of the fact that the eccentricities of the planets of the Solar system are actually quite small, with trajectories very close to a circle, which makes Kepler’s achievement (based on Tycho Brahe’s measurements) even more remarkable.
Here, I describe a simple measurement programme, suitable for first-year university students, consisting in measuring the elevation of the Sun as a function of time during a day, and in repeating this typically once a week over a full year. By measuring the maximal elevation hmax of the Sun, and the time tmax at which this maximum occurs (i.e. the true local noon), students can readily check that these quantities vary a lot over the year. The change in hmax is essentially related to the obliquity ε of the Earth over the ecliptic, and thus allows for quite an accurate determination of  ε (as well as that of the latitude of οbservation).
The change of tmax over a year gives an experimental determination of the equation of time E(t), i.e. the difference between the mean local noon and the true local noon, and allows for a determination of the eccentricity e of the Earth orbit [1]. This is a rewarding result for students to realize that with such simple measurements they can obtain good experimental values for the above quantities, and that with careful observations one can perform ‘science without instruments’ as did the astronomers of various antique civilizations [2, 3].
This article is organized as follows. I first describe how to measure in a simple way the elevation of the Sun versus time over a day, with an accuracy of about 1^o .
Then I give the results I obtained for hmax(t) and E(t) by repeating the measurement about once a week for one year, starting in August 2010. I show how one can extract the obliquity ε of the Earth’s axis and the eccentricity e of its orbit by fitting the experimental data with simple, analytic expressions. Finally, possible extensions of the work are proposed. Appendix A contains a brief reminder on basic notions of spherical astronomy, and should be read first by readers not familiar with these notions. In the remaining appendices, the derivation of the analytic expressions used for fitting the data is given, so that the article is self-contained…….

Read more: http://arxiv.org/pdf/1207.0982v1.pdf