The arbitrary division where short-period comets end and long-period comets begin is 200 years.

Using Kepler's third law for bodies orbiting the Sun, calculate the semimajor axes for the orbits of a short-period comet with a period of 151.0  years, and for a long-period comet of 751.0  years. 

To solve this, we will use **Kepler's Third Law**, which states that the square of the orbital period \( T \) of a body is proportional to the cube of the semi-major axis \( a \) of its orbit around the Sun. Mathematically, this is expressed as:

\[
T^2 = a^3
\]

where:
- \( T \) is the orbital period in years.
- \( a \) is the semi-major axis in astronomical units (AU).

### Step 1: Calculate the semi-major axis for the short-period comet (151 years)

Rearrange Kepler's Third Law:

\[
a = \sqrt[3]{T^2}
\]

For \( T = 151.0 \) years:

\[
a = \sqrt[3]{(151.0)^2}
\]

Let's calculate that.

### Step 2: Calculate the semi-major axis for the long-period comet (751 years)

Similarly, for \( T = 751.0 \) years:

\[
a = \sqrt[3]{(751.0)^2}
\]

I will now calculate both values.The semi-major axes for the orbits of the comets are:

- For the short-period comet with a period of 151.0 years, the semi-major axis is approximately **28.36 AU**.
- For the long-period comet with a period of 751.0 years, the semi-major axis is approximately **82.62 AU**.

These values show that the orbits of long-period comets are much more extended compared to short-period comets.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does Kepler's third law apply to planets within our solar system?
2. What is the difference between a comet's perihelion and aphelion?
3. How does the semi-major axis affect the shape of a comet's orbit?
4. What are the distinguishing characteristics of short-period versus long-period comets?
5. How would a comet's orbit change if its period were doubled?

**Tip:** The semi-major axis is a key factor in determining a comet's speed and how elliptical its orbit is—larger axes correspond to longer orbital paths and lower average speeds.

Using Kepler's third law for bodies orbiting the Sun, calculate the semimajor axes for the orbits of a short-period comet with a period of 151.0 years, and for a long-period comet of 751.0 years.

Learn how to calculate the semi-major axes of comets with orbital periods of 151.0 years and 751.0 years using Kepler's Third Law. This example demonstrates key concepts in astronomy and orbital mechanics, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation