Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Method of Eratosthenes. You are an astronomer on planet Nearth, which orbits a distant star. It has recently been accepted that Nearth is spherical in shape, though no one knows its size. One day, while studying in the library of Alectown, you learn that on the equinox your sun is directly overhead in the city of Nyene, located \(1000 \mathrm{km}\) due north of you. On the equinox, you go outside in Alectown and observe that the altitude of your sun is \(80^{\circ}\) What is the circumference of Nearth? (Hint: Apply the technique used by Eratosthenes to measure Earth's circumference.

Short answer

Nearth's circumference is calculated to be 36,000 km.

Step-by-step solution

Understanding the Problem

We are tasked with finding the circumference of the planet Nearth using the method of Eratosthenes. The city of Nyene is directly north of Alectown by 1000 km, and on the equinox, the sun is directly overhead in Nyene, meaning the sun makes a 0° angle with the vertical there. In Alectown, the sun's angle with the vertical is 80°.

Determine the Angular Difference

The angle formed by the sun's rays between Nyene and Alectown is crucial. In Nyene, the sun is directly overhead at 0°, and in Alectown, it is at 80°. Thus, the angular difference is \[ \theta = 90° - 80° = 10°. \]

Calculate the Circumference of Nearth Using Eratosthenes' Method

The formula applied by Eratosthenes is: \[ \frac{\text{Distance between two locations}}{\text{Circumference}} = \frac{\text{Angular difference}}{360°}. \]Plugging in the values, we have: \[ \frac{1000 \, \text{km}}{C} = \frac{10°}{360°}. \]Rearranging to solve for the circumference \(C\), we get: \[ C = \frac{1000 \, \text{km} \times 360°}{10°} = 36000 \, \text{km}. \]

Interpret the Result

Based on the calculation, the circumference of the planet Nearth is 36,000 km. This means that Nearth is a spherical planet with this circumference around its equator.

Key concepts

Eratosthenes' Method

Eratosthenes' Method is a classical approach used to measure the circumference of a spherical planet, such as Earth. Developed by the ancient Greek mathematician Eratosthenes around 240 BC, this method ingeniously applies simple geometry and observations to calculate the size of Earth without advanced technology.

The method primarily relies on the concept of angular differences between two locations along a meridian on the planet. By measuring the angle of the sun's rays at noon during an equinox, when the sun is directly overhead at one location, and comparing it to the angle at another location, one can infer valuable information.

A similar strategy was employed in the problem involving the imaginary planet Nearth, where the angular difference between two cities, Nyene and Alectown, was crucial to determining Nearth’s circumference. This method elegantly combines basic angular measurements with the distance between these two cities, demonstrating how ancient techniques remain relevant in understanding planetary dimensions today.

Spherical Planet Measurement

Measuring the dimensions of a spherical planet, like Nearth or Earth, requires a blend of observation and geometric reasoning. The fact that planets are approximately spherical in shape allows us to use unique mathematical techniques to calculate their size, such as circumference.

The core principle lies in understanding how angles relate to the planet's surface. When we know the distance between two points and the angle difference of sunlight striking them, we can extrapolate these measurements to express the whole circumference.

On Nearth, the distance of 1000 kilometers between two cities and a 10° angle difference were pivotal in deriving the planet's circumference using Eratosthenes’ formula. This formula essentially states:

• The ratio of the distance between the two points to the total circumference is equal to the ratio of the angular difference to a full 360°.

This straightforward equation relies on the understanding of the circle's geometry and assures that even with minimal data, robust measurements can be achieved.

Equinox Observations

Equinox observations are critically useful in determining planetary measurements because of the unique position of the sun relative to the Earth. An equinox occurs twice a year, marking the moments when the sun is directly above the equator, leading to nearly equal day and night length globally.

During these times, the sun's rays strike the equator perpendicularly. This makes calculations more simplified as one of the locations involved will have the sun directly overhead, creating a vertical angle of 0°.

In the Nearth problem, the equinox observation enabled astronomers to gather clear data about the sun's position above Nyene, setting the stage to understand the planetary scale. The observation was essential because it gave a clear reference point from which angular differences at other latitudes could be measured without complex adjustments for the sun's varying position across different times of the year. This direct linkage to solar positioning during equinox periods remains a foundational aspect of celestial and terrestrial measurements.