Problem 2 - Astronomy Journey to the Cosmic Frontier

Question

Describe a way in which Eudoxus's model could have explained the motion of the Sun, both during a day and throughout a year.

Step-by-Step Solution

Answer

Eudoxus used two rotating spheres: one for daily Sun motion and another for seasonal changes over a year.

Step 1: Understand Eudoxus's Model

Eudoxus of Cnidus proposed a geocentric model with concentric spheres to explain celestial motions. For the Sun, this involves imagining it attached to a sphere that rotates around the Earth.

Step 2: Daily Motion of the Sun

In Eudoxus's model, the Sun is attached to a sphere that completes one full rotation around the Earth every 24 hours. This explains the Sun's daily path as it rises in the east and sets in the west.

Step 3: Annual Motion of the Sun

For the yearly motion, Eudoxus added a second sphere that causes slight variations in the main sphere's rotation axis. This second sphere rotates once a year, explaining the Sun's apparent path change with seasons.

Step 4: Combining Spheres for Motion Explanation

The combination of the two spheres—one for daily motion and one for annual motion—results in the observed daily arcs and seasonal shifts of the Sun's path across the sky.

Key Concepts

Geocentric Model

In ancient Greek astronomy, the geocentric model was a pivotal concept. It proposed that the Earth was at the center of the universe. This idea was rooted in the observation that celestial bodies, like the Sun and stars, appeared to move around the Earth. Eudoxus of Cnidus, a Greek astronomer, supported this model. He suggested that heavenly bodies were embedded in spherical shells, each concentric with the Earth. This meant every sphere had the same center as Earth.
One of the main features of Eudoxus's geocentric model was its ability to explain the complex motions of celestial bodies. It was incredibly intuitive, as people could see everything rotating around them. Despite its elegance, it would later be replaced by the heliocentric model, which proved more accurate in explaining celestial phenomena.

Concentric Spheres

Eudoxus's model relied on concentric spheres to explain the movement of celestial bodies. These spheres were imagined not as solid objects we could see but as theoretical constructs. Each celestial body, including the Sun, was attached to an invisible sphere.

• Each sphere shared the same center, the Earth.
• The outermost sphere was thought to contain the stars, all rotating around the center Earth.

This layering of spheres allowed for complex motion patterns. For example, the Sun's motion involved more than one sphere. This model's simplicity was its genius, making it easy for people to grasp the plotted paths of celestial bodies.

Celestial Motions

Celestial motions in Eudoxus's model were explained through the rotation of the concentric spheres. As each sphere rotated, it carried its attached celestial body along a specific path.
The spheres moved differently:

• Some completed rotations within one day.
• Others took longer, such as a year, to complete a cycle.

By varying the rotation speed and axis tilt of these spheres, Eudoxus's model could mimic the observable changes in the sky. This complex yet methodically structured system tried to account for the intricate dance of planets, stars, and the Sun as seen from Earth.

Sun's Daily Path

The Sun's daily path according to Eudoxus was mapped by a single sphere. This sphere completed a full rotation around the Earth every 24 hours.
As the sphere rotated, it resulted in the visible rise and fall of the Sun from east to west. This mirrored our modern understanding of the Earth's rotation, although Eudoxus's model kept Earth stationary. The daily revolution explained:

• The regular cycle of day and night.
• The consistent pattern of the Sun's movement across the sky.

Although simplistic by today’s standards, Eudoxus's explanation of the Sun's daily path was a significant step in early astronomical modeling.

Sun's Annual Motion

Explaining the Sun's annual motion involved more than just a daily rotating sphere. Eudoxus added a second sphere to account for changes across the year. This secondary sphere rotated at a slower pace:

• One complete rotation took one year.
• This created a slight variation in the axis of the main daily sphere.

As a result, the Sun appeared to travel along different paths depending on the season. This accurately reflected the change in the Sun's altitude and length of days over the year. The addition of an annual sphere made it possible to explain seasonal changes, such as:

• The differing lengths of daylight.
• The changing position of the Sun at noon over months.

Eudoxus's multi-sphere solution offered a clever attempt to explain the observable celestial behaviors linked to the Sun's annual journey across the sky.