Explain why the semimajor axis of a planet's orbit is equal to the average of the distance from the Sun to the planet at perihelion (the perihelion distance) and the distance from the Sun to the planet at aphelion (the aphelion distance).

Imagine journeying around the sun on a path that sometimes brings you closer and sometimes takes you farther away. That's exactly the adventure a planet embarks upon in its orbit. The closest point in this journey is called the perihelion, and it is where a planet enjoys the warmest embrace of the sun. In contrast, the aphelion is like the planet's vacation away from the sun’s heat, marking the furthest point in the orbit.

Mathematically speaking, the perihelion distance, denoted as \( r_p \), and the aphelion distance, \( r_a \), are not just points but crucial metrics that astronomers use to calculate a variety of orbital characteristics, including the semimajor axis. Understanding these distances helps us piece together the cosmic dance that planets perform around their star, and it’s key to unraveling many of the mysteries of their motion.

A journey through space is seldom a perfect circle. In the vast expanses of the cosmos, planets follow a path that stretches more in one direction, creating an ellipse. It's akin to stretching out a rubber band—while it retains its closed loop, its shape is elongated. Elliptical orbits showcase this natural phenomenon in the orbits of planets.

The semimajor axis traces through the longest part of this elongated path, making it a fundamental parameter in describing an orbit’s size and shape. Think of it as the heart within the heart that holds constant, even as the planet moves closer or farther from the sun. By grasping the concept of elliptical orbits, we take a significant step in the vivid portrayal of planetary movement.

Astronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere. It's a field that encompasses the grand endeavor to comprehend the universe and our place within it, brought into focus through the lens of telescopes and the insights of theoretical principles.

In studying celestial mechanics, such as the orbits of planets, astronomy combines observations with laws of physics to describe the harmonious yet complex ballet of the heavens. The distances from a planet to the sun at various points in its orbit are observable facts that astronomers rely on to infer deeper truths about the forces and events that shape our universe.

To unlock the celestial secrets of planetary movement, one must turn the pages of history to Johannes Kepler's contributions. His three laws of planetary motion serve as the bedrock of celestial mechanics. They tell us that planets orbit in ellipses (First Law), sweep out equal areas in equal times (Second Law), and have square of their orbital periods proportional to the cube of the semimajor axes of their orbits (Third Law).

Kepler’s elegant laws bring reason to the observed patterns of movement among heavenly bodies. They explain that while planets may vary their speed and distance from their star, there is a consistency—much like there is in music or poetry. It is through these laws that the calculation of the semimajor axis from perihelion and aphelion distances becomes more than mere arithmetic; it is, in fact, a poetic quantification of the celestial order.