Earth radius from a single sunrise image: a classroom-ready activity

This paper presents a classroom activity where students estimate Earth's radius by analyzing a sunrise photograph of Mont Blanc's shadow from Geneva, using geometric reasoning and atmospheric refraction corrections to derive an upper bound that serves as a practical lesson in scientific modeling and the Nature of Science.

The Gist

Imagine you are standing in Geneva, looking out at the sunrise. The sun is just peeking over the horizon, but it's so low that it hasn't actually cleared the curve of the Earth yet. However, its light is strong enough to shine over the top of a distant mountain, Mont Blanc, and cast a giant shadow onto a layer of clouds floating above that mountain.

This paper is essentially a clever "detective story" where two teachers use that single photograph to solve a mystery: How big is the Earth?

Here is the story of how they did it, broken down into simple steps:

1. The Clue: A Shadow on a Cloud

Usually, to measure the Earth, you need to travel thousands of miles to different cities (like the ancient Greek Eratosthenes did). But these authors found a shortcut. They took a photo of Mont Blanc's shadow hitting a cloud layer.

Think of the sunlight as a giant, invisible ruler. Because the sun is so far away, its rays are almost perfectly parallel. When those rays hit the top of Mont Blanc and continue on to hit the cloud, they create a specific angle. By measuring exactly where the shadow falls on the cloud compared to the mountain peak, they could figure out the angle at which the sun's light was hitting the mountain.

2. The Camera as a Protractor

The photo wasn't just a pretty picture; it was a measurement tool. The authors used the camera's lens like a protractor.

• They measured the distance between the mountain peak and its shadow in "pixels" on the photo.
• They knew the size of the camera's sensor and the lens's focal length (the "zoom" power).
• Using basic math (trigonometry), they converted those pixels into a real-world angle. It's like knowing that if a shadow is a certain length on a wall, the light source must be at a specific angle.

They calculated that the sun's rays were hitting the mountain at an angle of about 88.9 degrees relative to straight up. (If the Earth were flat, the angle would be exactly 90 degrees. The fact that it's slightly less proves the Earth is curved).

3. The "Flat Earth" Test

To find the Earth's radius, they imagined a simple geometric triangle:

• Point A: The center of the Earth.
• Point B: The top of Mont Blanc.
• Point C: The point where the sun's ray just grazes the Earth's surface (tangent) before hitting the mountain.

If the Earth were flat, the light would hit the mountain straight on. Because the Earth is round, the light has to "dip" slightly to clear the curve. The steeper the dip, the smaller the Earth; the shallower the dip, the larger the Earth.

Using their calculated angle, they did the math and got a result: The Earth's radius is about 26,600 km.

4. The Reality Check: The "Atmospheric Lens"

Here is where the story gets interesting. The actual Earth radius is only about 6,370 km. Their first guess was more than four times too big!

Why? They realized they forgot about the atmosphere.
Think of the Earth's atmosphere like a giant, curved lens. As light travels through the air, the air gets thinner as you go up. This bends the light rays downward, just like a straw looks bent in a glass of water.

• The Effect: This bending makes the sun look higher in the sky than it actually is.
• The Correction: The authors adjusted their math to account for this "bending" (refraction). They subtracted a tiny bit from their angle (about 0.6 degrees).

5. The Final Result

After correcting for the atmosphere, their new estimate for the Earth's radius dropped to about 10,900 km.

• This is still about 1.7 times larger than the real Earth.
• But, considering they only used a single photo, a camera, and some high-school math, getting within a factor of two is a huge success!

Why This Matters for Students

The authors aren't just trying to prove the Earth is round (we already know that). The real goal of this paper is to show students how science works:

1. Observation: Look at a simple, everyday thing (a shadow).
2. Modeling: Build a mathematical model to explain it.
3. Error Analysis: Realize your first answer is wrong because you missed a factor (the atmosphere).
4. Refinement: Fix the model and get a better answer.

It teaches that science isn't about getting the perfect number immediately; it's about understanding why your numbers are off and how to improve your model. It turns a simple sunrise photo into a lesson on geometry, physics, and the scientific method.

Technical Summary

1. Problem Statement

The paper addresses the challenge of estimating the Earth's radius using only local observations and a single photograph, avoiding the need for large-scale geodetic surveys or widely separated observation points (unlike historical methods by Eratosthenes or Al-Biruni). Specifically, the authors aim to derive a conservative upper bound on the Earth's radius using a photograph taken in Geneva showing the shadow of Mont Blanc projected onto a cloud layer. The study also serves as a pedagogical tool to demonstrate the scientific method, including hypothesis formulation, modeling, and uncertainty analysis, for upper-secondary STEM students.

2. Methodology

The methodology combines observational astronomy, geometric optics, trigonometry, and atmospheric physics. The process is divided into four main stages:

A. Data Extraction and Image Geometry

• Observation: A photograph was taken in Geneva (altitude $h_G = 430$ m) on January 7, 2022, at 08:06 local time. It captures the shadows of Mont Blanc ($h_B = 4810$ m) and Mont Maudit ($h_M = 4465$ m) cast onto a stratus cloud layer.
• Pixel Measurement: The authors measured pixel separations on the cropped image:
  • Radial separation between the two shadows: $\Delta d_{pix} = 307$ px.
  • Radial separation between Mont Blanc and its shadow: $x_{pix} = 105$ px.
  • Total image height: $4608$ px.
• Optical Conversion: Using the camera's sensor dimensions ($18.0 \text{ mm} \times 13.5 \text{ mm}$) and focal length ($f = 42.0$ mm), pixel distances were converted to physical distances on the sensor and then to angular separations ($\delta$ and $\theta$) using the lens equation (assuming the object distance is much larger than the focal length):
  • Angular separation between shadows: $\delta = 1.64^\circ$.
  • Angular separation between Mont Blanc and its shadow: $\theta = 0.560^\circ$.

B. Geometric Modeling (Determining the Angle of Light)

The authors constructed a geometric model to determine the angle of incidence of the solar rays ($\alpha$) relative to the local vertical at Mont Blanc's summit.

• Knowns: Elevation differences, horizontal distance to Mont Blanc ($d = 71$ km), and the measured angles $\delta$ and $\theta$.
• Unknowns: The height of the cloud layer above the summit ($x$) and horizontal distances to the shadows.
• Calculation: By applying right-triangle trigonometry and solving a system of equations based on similar triangles formed by the parallel solar rays, the authors solved for the cloud height ($x \approx 146$ m).
• Result: The angle of light $\alpha$ was calculated as $88.9^\circ$ relative to the local vertical.

C. Radius Estimation (Geometric Limit)

Using Al-Biruni's method, the authors assumed the Earth is a perfect sphere and that the sunlight casting the shadow is tangent to the Earth's surface.

• Formula: $R = h_B \cdot \frac{\sin(\alpha)}{1 - \sin(\alpha)}$
• Initial Result: Substituting the values yielded an upper bound of $R_{max} \approx 26,600$ km.
• Note: This is significantly larger than the accepted Earth radius ($\approx 6,371$ km), indicating the need for corrections.

D. Refraction Correction

The authors refined the estimate by accounting for atmospheric refraction, which bends light rays downward as they pass through the atmosphere, making the Sun appear higher than its geometric position.

• Correction: A standard horizon refraction correction of $34'$ (arcminutes) was applied to $\alpha$.
• Corrected Angle: $\alpha_{corr} = 88.9^\circ - 34' = 88.3^\circ$.
• Corrected Result: The revised upper bound became $R_{max, corr} \approx 10,900$ km.

3. Key Contributions

• Pedagogical Framework: The paper provides a structured, classroom-ready activity that guides students from simple observation to complex modeling. It emphasizes the "Nature of Science" (NOS) by requiring students to formulate hypotheses, handle approximations, and analyze sources of error.
• Single-Image Methodology: It demonstrates that a single photograph, combined with known geographic data and basic trigonometry, can yield a physically meaningful (albeit an upper-bound) estimate of planetary scale.
• Error Analysis: The study explicitly identifies and quantifies the sources of overestimation:
  1. Atmospheric Refraction: The primary source of error, addressed in Section 5.
  2. Topography: The assumption of a perfectly spherical Earth ignores local terrain variations that might block the tangent ray.
  3. Mathematical Sensitivity: The function relating the angle $\alpha$ to the radius $R$ is highly non-linear near $90^\circ$. A small error in $\alpha$ (e.g., 1%) leads to a massive error in $R$ (>10%), illustrating the importance of precision in limiting cases.

4. Results

• Initial Estimate: $26,600$ km ($\approx 4.2 \times$ the actual Earth radius).
• Refraction-Corrected Estimate: $10,900$ km ($\approx 1.7 \times$ the actual Earth radius).
• Cloud Height Verification: The calculated cloud height ($\approx 4956$ m) was cross-validated with meteorological data from MétéoSuisse, confirming the model's internal consistency.
• Sensitivity: The study confirms that while the geometric derivation is robust, the final radius estimate is extremely sensitive to the precise value of the solar elevation angle due to the asymptotic nature of the tangent function near the horizon.

5. Significance

• Educational Value: The activity bridges the gap between abstract physics concepts (trigonometry, optics, refraction) and real-world observation. It is accessible to upper-secondary students (high school level) yet rigorous enough to illustrate complex scientific reasoning.
• Scientific Method Illustration: It concretely demonstrates how scientific models are iterative. The initial model (geometric only) yields a "wrong" but bounded result; the refined model (including refraction) improves accuracy; and the discussion of limitations (topography, sensitivity) teaches critical evaluation.
• Accessibility: It proves that significant scientific inquiry does not always require advanced technology or global datasets; careful observation of everyday phenomena (sunrise shadows) can yield profound insights into planetary geometry.

In conclusion, the paper successfully uses a specific photographic event to derive a conservative upper bound for the Earth's radius, transforming a simple image into a comprehensive lesson on geometric modeling, atmospheric physics, and the limitations inherent in scientific measurement.