This paper documents a hands-on experiment to estimate the diameter of the Sun by using a flat mirror with a small pinhole aperture to project a solar image onto a wall. The student sets the mirror-to-wall distance to approximately six meters and records the diameter of the reflected solar image at three ten-minute intervals. Applying a simple ratio-and-proportion formula — D/L = d/l, where L is Earth's mean orbital radius (150,000,000 km) — each measurement yields an estimate of the Sun's diameter, and the three results are averaged to produce a final value of approximately 1,404,502.39 km. The paper also reflects on experimental error, scientific discipline, and the broader significance of applying mathematics to astronomy.
Measuring the diameter of the Sun relative to the average radius of Earth's orbit requires basic geometric knowledge, particularly an understanding of angle properties. Grasping these properties reveals the mathematical relationship between the data provided in the problem and the data collected during the experiment. The mathematical aspect, however, is only half of the story — reckless experimentation produces errors that are difficult to tolerate.
The choice of favorable conditions, the careful arrangement of materials, and the precise recording of measurements all contribute to obtaining a reliable result. These three factors are nevertheless subject to some degree of error no matter how careful the experimenter may be. Human error is always present, and every instrument has limitations that affect the final result. The precision of the materials used and of the measurements taken ultimately determines how precise the result will be.
In conducting this experiment, I carefully set up the required materials. I used a flat mirror covered by a sheet of white paper with a small hole in the middle, which I measured to be approximately 7 millimeters in diameter. Finding a suitable location was relatively straightforward; the harder task was angling the mirror so that the Sun's image projected onto the beige-colored wall appeared as close to a circle as possible — not oblong or elliptical — and setting the mirror-to-image distance at almost exactly 6 meters. Setup was complete at 3:07 in the afternoon. Beginning at 3:10 pm, I took the required measurements at ten-minute intervals.
Approaching the problem mathematically, I first identified what the experiment required, what data were already given, what variables needed to be measured, and what formulas applied. This step-by-step procedure is the key to obtaining the desired result.
The required quantity is D, the diameter of the Sun. The given datum is L, the average radius of Earth's orbit, equal to 150,000,000 km. The quantities to be measured directly are: l, the distance from the mirror to the reflected image on the wall, and d, the diameter of that image. The governing relationship is a simple ratio and proportion:
D : L = d : l, or equivalently, D / L = d / l.
Solving for D gives the working formula: D = (d / l) × L. All three measurements of d and l are substituted into this formula to calculate the corresponding estimates of D.
The table below shows the values of d and l recorded at each measurement interval, along with the resulting estimate of D and the final average.
For the first measurement, taken at 3:10 pm, d = 55.7 mm and l = 5,998 mm. Substituting into the formula: D = (55.7 / 5,998) × 150,000,000 km = 1,392,964.32 km.
For the second measurement, taken at 3:20 pm, d = 56.25 mm and l = 6,000 mm. By substitution: D = (56.25 / 6,000) × 150,000,000 km = 1,406,250.00 km.
For the third measurement, taken at 3:30 pm, d = 56.6 mm and l = 6,003 mm. By substitution: D = (56.6 / 6,003) × 150,000,000 km = 1,414,292.85 km.
"Averages three D values to ~1,404,502 km"
"Reflects on scientific discipline and mathematical wonder"
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