A comet orbits the Sun with a semimajor axis of 340.7 AU. Using Kepler's third law, the comet's orbital period is approximately how many years?

Kepler's third law relates the orbital period \( T \) of a body orbiting the Sun to the semimajor axis \( a \) of its orbit. The law can be expressed as:

\[
T^2 = a^3
\]

where:
- \( T \) is the orbital period in Earth years,
- \( a \) is the semimajor axis in astronomical units (AU).

Given that the semimajor axis of the comet's orbit is 340.7 AU, we can substitute this value into the equation:

\[
T^2 = (340.7)^3
\]

Now, solving for \( T \):

\[
T = \sqrt{(340.7)^3}
\]

Let's calculate that.The comet's orbital period is approximately 6,289 years.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How does Kepler's third law apply to planets in the Solar System?
2. What is the significance of the semimajor axis in orbital mechanics?
3. How does a comet's elliptical orbit affect its speed at different points?
4. How would the orbital period change if the semimajor axis were halved?
5. What is the difference between the sidereal period and synodic period in celestial mechanics?

**Tip:** The orbital period grows rapidly as the semimajor axis increases, due to the cubic relationship in Kepler's third law.

Explore how to calculate the orbital period of a comet with a semimajor axis of 340.7 AU using Kepler's third law. This problem delves into orbital mechanics and astronomy, providing a step-by-step solution. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation