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.