When a person or object moves further away from an observer, it eventually disappears from sight. This disappearance can happen for two primary reasons:
Option 1) Angular Resolution and Perspective:
In this situation, the ground is effectively “brought” to your eye level as the object moves away. The person or object appears smaller with distance until it vanishes due to the angular resolution limit of the human eye. The angular resolution limit defines the smallest angle at which two objects can be distinguished as separate. For the human eye, this limit is approximately 0.0216 degrees.
Option 2) Physical Curvature
The disappearance is attributed to the curvature of the Earth. As the object moves farther away, it eventually passes over the horizon and is no longer visible. In this scenario, the observer’s eyeline must continually shift downward to keep sight of the object, but eventually, the curvature of the Earth causes the object to disappear from view.
To estimate the Earth’s radius using the angular resolution limit of the human eye, we can make use of some basic geometry and the information provided in the second option. We assume a person standing at 6 feet (or approximately 1.8288 meters) tall is walking away, and their disappearance is a result of moving over the curvature of the Earth.
The angular resolution limit of the human eye is taken to be 0.0216 degrees. This small angle corresponds to the point at which the person can no longer be distinguished by the eye and disappears from view. By combining this with known geometric properties of circles, we can estimate the radius of the Earth.
The formula used to calculate the distance to the horizon for an object of height h over a sphere with radius R (the Earth in this case) is:
d = sqrt(2Rh + h^2)
Where:
d is the distance to the horizon, or the point where the person disappears,
h is the height of the observer (1.8288 meters),
R is the radius of the Earth.
For small angles, we can also use a tangent approximation to estimate the radius based on the angular resolution:
R = h / [2 * (tan(theta))^2]
Where:
theta is the angular resolution limit, which is 0.0216 degrees.
By plugging in these values, we calculate the radius of the Earth to be approximately 6433.9 kilometers.
The calculated radius of 6433.9 kilometers is a reasonable estimate, though it is slightly larger than the average accepted radius of the Earth, which is about 6371 kilometers. This difference may arise from simplifications and assumptions in the model, such as assuming perfect conditions for visibility and a perfectly spherical Earth. Nonetheless, this calculation demonstrates how we can use the angular resolution of the human eye to arrive at an estimate of the Earth’s curvature and radius.
Source: “Angular Resolution and Our World” 3hr Lecture