What is the (fictitious) mean sun? What path does it follow on the celestial sphere? Why is it a better timekeeper than the actual Sun in the sky?

The celestial equator is a key concept in astronomy, denoting an imaginary circle that perfectly mirrors Earth's equator but projected onto the celestial sphere. This makes it an essential element in mapping the skies and understanding the apparent paths that celestial bodies take as observed from Earth.

Think of the celestial equator as an invisible ring floating in space, aligned with the equator on our planet. Celestial bodies appear to move above and below this invisible line, which is important for astronomers when tracking movements across the sky. Because the equator represents the midpoint between the northern and southern hemispheres, the celestial equator is also evenly split between the celestial northern and southern hemispheres.

The concept of the celestial sphere provides us with a mental model to visualize how the heavens are organized from our viewpoint. Imagine a vast, hollow sphere surrounding the Earth with the stars and other celestial bodies embedded on its inner surface. This is the celestial sphere, and even though it isn't a physical entity, this model helps us comprehend the positions and movements of celestial objects.

Any point above the Earth’s surface can be projected outward to this sphere, so we have poles aligned with Earth's poles and the aforementioned celestial equator, which finds its counterpart in Earth's own equator. As Earth rotates and orbits the Sun, celestial bodies appear to travel across this sphere, creating a dynamic celestial map that has guided explorers, scientists, and timekeepers for centuries.

Solar timekeeping is rooted in the apparent motion of the Sun across our sky. It is centered around the concept of a 'solar day,' which represents the time it takes for the Sun to return to the same position in the sky, such as from one high noon to the next. However, due to the elliptical nature of Earth's orbit and its axial tilt, the actual Sun can appear to speed up or slow down throughout the year.

This is where the mean sun, a hypothetical perfect timekeeper, comes into play. By averaging the apparent motion of the actual Sun over the year, the mean sun moves uniformly along the celestial equator, providing a consistent measure for a solar day. This consistency is crucial for accurate timekeeping, as it remains steady despite the eccentricities of the actual Sun's apparent journey in our sky.

The apparent motion of the Sun is the visible path it seems to take when viewed from Earth. This motion is primarily a result of Earth's rotation on its axis and orbit around the Sun, leading observers to see the Sun rise in the eastern sky, arc across the heavens, and set in the west.

The Sun doesn't move at a uniform rate in the sky; it varies due to the tilt of Earth’s axis and its elliptical orbit. Sometimes the Sun appears to move faster, such as during perihelion (closest point to the Sun), and slower during aphelion (farthest point from the Sun). The inclination of Earth also results in the Sun taking a higher or lower path across the sky, contributing to the seasons. For practical purposes, the mean sun's consistent path along the celestial equator simplifies this intricate dance into a steady rhythm for timekeeping.