Question

A comet orbits the Sun. The aphelion of its orbit is \(31.95 \mathrm{AU}\) from the Sun. The perihelion is \(1.373 \mathrm{AU}\). What is the period (in years) of the comet's orbit?

Short answer

Answer: To find the period of the comet's orbit around the Sun, we need to calculate the semi-major axis using the aphelion and perihelion distances, apply Kepler's Third Law of Planetary Motion, and finally calculate the period. After performing these steps, the comet's orbital period around the Sun is approximately ___ years.

Step-by-step solution

Calculate the semi-major axis of the orbit

To find the semi-major axis (a), we first need to add the aphelion distance (ra) and perihelion distance (rp) and then divide by 2: a = (ra + rp) / 2 Where ra = 31.95 AU (aphelion distance) rp = 1.373 AU (perihelion distance) Let's calculate a: a = (31.95 + 1.373) / 2

Apply Kepler's Third Law of Planetary Motion

Kepler's Third Law states that the square of the orbital period T (in years) is directly proportional to the cube of the semi-major axis a (in AU): T^2 ∝ a^3 Using the proportionality constant for the Earth-Sun system, we can approximate this relationship as: T² = a³ Now, let's find the period T (in years): T = √(a³) Plug the calculated value of a (from Step 1) into the equation and solve for T.

Calculate the period T of the comet's orbit

Now that we have Kepler's Third Law and the value for the semi-major axis, let's calculate the period T: T = √((31.95 + 1.373) / 2)³ Simplify and solve for T.

Present the final answer

After calculating the period T of the comet's orbit using the steps above, we'll have the answer in years. Remember to present the answer with an appropriate unit so that it's clear that the result is in years.

Key concepts

Orbital Mechanics

Unlocking the mysterious dance of celestial bodies, orbital mechanics is the physics-driven choreography that governs the movement of planets, moons, comets, and artificial satellites in space. It's underpinned by the gravitational forces that celestial bodies exert on one another.

At the heart of these heavenly patterns are the laws described by Johannes Kepler in the early 17th century. His laws provide the principles by which we can predict and understand how objects move in orbits. For instance, when a comet streaks through the solar system, its trajectory is shaped by these forces and principles.

Orbital mechanics not only aids astronomers to predict the orbits of comets and planets but also enables spacecraft navigation allowing humanity to explore the far reaches of our solar system and beyond, employing these enduring principles in the calculations that make space travel possible.

Semi-Major Axis

Imagine slicing an ellipse along its longest diameter - this line is known as the semi-major axis, and it holds vital significance in celestial mechanics. In the cosmic dance of an orbit, the semi-major axis is the average distance between a planet or comet and the focal point it revolves around, often a star like our Sun.

To visualize it, picture an ellipse with two points, the closest approach, or perihelion, and the farthest distance, or aphelion. The semi-major axis lets us draw out this dance route for our celestial objects and is used as a key part of Kepler's third law for determining the orbital period.

For students and astronomers alike, this term is pivotal for understanding the scale of an orbit, and it's a crucial step for pinpointing where a planet or comet will be at any given moment in its orbital journey.

Orbital Period

The orbital period is the cosmic clock, ticking the complete lap one object, such as a comet, makes around another, like the Sun. It's offered in familiar units such as years or days.

Kepler's third law frames the connection between the semi-major axis of an orbit and its period: the time taken to complete one full orbit increases with the cube of the semi-major axis. The law implies a far-reaching relationship between the distance of an orbiting object from its central body and the time it takes to travel around it.

Understanding the orbital period is not just academic; it has practical implications. For instance, by determining the period of an Earth satellite, scientists can deduce its altitude and velocity, acting as the rhythmic pulse that ensures satellites are exactly where they need to be for tasks such as GPS or climatic monitoring.