Question

The object Sedna orbits our Sun at an average distance (semimajor axis) of 509 AU. What is its orbital period?

Short answer

Sedna's orbital period is approximately 11,446 years.

Step-by-step solution

Understand Kepler's Third Law

Kepler's Third Law of planetary motion states that the square of the orbital period $T$ of a planet is directly proportional to the cube of the semi-major axis $a$ of its orbit. This can be expressed as $T^2=a^3$ when using astronomical units (AU) for distance and years for time.

Prepare the Equation for Sedna's Orbit

We know Sedna's semimajor axis is $509$ AU. According to Kepler's Third Law, $T^2={509}^3$. Our task is to now solve this equation to find $T$, Sedna's orbital period.

Calculate ${509}^3$

First, compute ${509}^3$. This results in $509\times 509\times 509$. By doing this calculation, we find that ${509}^3=131,045,509$.

Solve for T

Using the relation from Kepler's Third Law, $T^2=131,045,509$. To find $T$, we take the square root of $131,045,509$: $T=\sqrt{131,045,509}$.

Calculate the Square Root

Compute $\sqrt{131,045,509}$. This calculation results in approximately $11,446$ years. Therefore, the orbital period $T$ of Sedna is roughly 11,446 years.

Key concepts

Astronomical Units

Astronomical Units (AU) are a convenient way to express distances within our solar system. One astronomical unit is the average distance from the Earth to the Sun, approximately 93 million miles or 150 million kilometers. This unit simplifies comparisons and calculations of vast cosmic distances, making it easier to grasp the scale of our solar system. For instance, Earth's orbit is about 1 AU from the Sun, while Sedna orbits at an average of 509 AU, highlighting just how far it ventures into the outer reaches of the solar system.

By using AUs, calculations become manageable, especially when applying Kepler's Third Law to determine the orbital periods of celestial objects. It allows astronomers to easily relate massive distances with something more comprehensible and standardized for solar system measurements.

Orbital Period

The orbital period represents the time it takes an astronomical object to complete one full orbit around the Sun. For planets and other celestial bodies, this period is often measured in Earth years. Understanding the concept of orbital periods is crucial in studying planetary movement and predicting future positions within the solar system.

In our example, Kepler's Third Law assists in determining Sedna's orbital period. By understanding that the square of the orbital period is proportional to the cube of the semimajor axis, we can calculate how long it takes Sedna to make one complete journey around the Sun. After solving, we found that Sedna's orbital period is approximately 11,446 years, indicating that it has a very long and distant orbit.

Solar System Dynamics

Solar System Dynamics involves the study of gravitational interactions and motions of celestial bodies within our solar system. These dynamics are governed by a set of laws and principles introduced by astronomers like Johannes Kepler and Isaac Newton.

Kepler's Laws of Planetary Motion, including the Third Law, help explain and predict the behaviors of objects such as planets, dwarf planets, comets, and asteroids. They provide a crucial framework for understanding orbit shapes, sizes, and durations. Learning about these dynamics sheds light on how forces like gravity ensure that everything from tiny asteroids to massive planets move in predictable paths.

Understanding how Sedna fits into this framework, with its distant and lengthy orbit, broadens our knowledge of the solar system's structure and the gravitational influences at play, especially in the far reaches beyond the traditional planets.