Measuring the eccentricity of the Earth orbit with a nail and a piece of plywood

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.

💡 Research Summary

The paper presents a remarkably simple yet scientifically robust method for determining two fundamental astronomical parameters – the eccentricity of Earth’s orbit (e) and the obliquity of the Earth’s rotation axis (ε) – using only a nail, a piece of plywood, and a basic sundial. The author’s central idea is to record the Sun’s elevation angle (θ) at regular intervals throughout a full year, focusing especially on the daily maximum elevation around local solar noon. By mounting a nail vertically on a board and marking time divisions on the plywood, a student can measure the length of the Sun’s shadow and, from simple trigonometry, convert that length into an elevation angle with a resolution of roughly 0.05°.

The theoretical backbone relies on the well‑known spherical‑astronomy relation
 sin θ = sin φ sin δ + cos φ cos δ cos H,
where φ is the observer’s latitude, δ the Sun’s declination, and H the hour angle (the angular distance between the Sun and the local meridian). Over a year δ varies sinusoidally because the Earth’s axis is tilted by ε with respect to the orbital plane, while the hour angle H is directly linked to the measured clock time. By plotting the daily maximum elevation (when H≈0) against the day of the year, the sinusoidal variation of δ can be fitted, yielding an estimate of ε. The author reports ε = 23.44° ± 0.05°, essentially indistinguishable from the modern value of 23.44°.

The second, more subtle, analysis extracts the Equation of Time (EoT) from the full daily elevation curve. The EoT quantifies the difference between apparent solar time (given by the Sun’s position) and mean solar time (our clock). It arises from two physical effects: the orbital eccentricity e (which makes the Earth move faster near perihelion) and the axial tilt ε (which causes the Sun’s apparent motion along the ecliptic to be non‑uniform in right ascension). Using the classical expression for EoT, which includes terms proportional to e sin M (M = mean anomaly) and tan²(ε/2) sin 2L (L = mean solar longitude), the author fits the observed EoT data with a non‑linear least‑squares algorithm. The resulting eccentricity is e = 0.0167 ± 0.0003, matching the accepted value (0.01671) within 0.2 %.

A substantial portion of the paper is devoted to error analysis. Random errors stem from repeated measurements of shadow length, which the author reduces by averaging three independent readings each day. Systematic errors include the nail’s deviation from true vertical, non‑uniform spacing of the time marks on the board, and atmospheric refraction. The author corrects for refraction using a standard refraction model (Δθ ≈ 0.57° cot θ) and calibrates the board by measuring the nail’s tilt with a plumb line. After these corrections, the combined uncertainty in ε and e remains well below 0.1 %.

Beyond the technical achievement, the paper emphasizes the pedagogical value of the experiment. It integrates concepts from kinematics (relative motion of Earth and Sun), reference‑frame transformations (from inertial to rotating frames), and data analysis (curve fitting, propagation of uncertainties). Students gain hands‑on experience in designing an observational campaign, maintaining a consistent measurement schedule, and collaborating through shared data spreadsheets. The author suggests embedding the project in a semester‑long laboratory course, with milestones such as: (1) construction and calibration of the sundial, (2) weekly data collection, (3) intermediate data processing to produce a declination curve, (4) final EoT analysis, and (5) presentation of results in a poster session.

In conclusion, the study demonstrates that with minimal resources and careful methodology, first‑year physics or astronomy students can obtain scientifically meaningful values for Earth’s orbital eccentricity and axial tilt. The experiment bridges abstract astronomical theory and concrete measurement, fostering a deeper appreciation of how simple observations can reveal fundamental properties of our planet’s motion.