1.) Halley’s comet has an observed period of 76.3 years. Use Kepler’s 3rd law to calculate the semimajor axis of the comet’s orbit2.) The dwarf planet Makemake is located a distance 45.8 AU from the sun. What is its orbital period in years?

Answer 1

1. Semimajor Axis of Halley’s Comet

Kepler’s 3rd law states:

P^2 = a^3

where P is the orbital period in years, and a is the semimajor axis in astronomical units (AU).

Given:

P = 76.3 \text{ years}

We need to solve for a :

a^3 = P^2

a^3 = (76.3)^2

a^3 = 5821.69

a = \sqrt[3]{5821.69}

a \approx 17.94 \text{ AU}

So, the semimajor axis of Halley’s comet’s orbit is approximately 17.94 AU.

2. Orbital Period of Makemake

Given:

a = 45.8 \text{ AU}

Using Kepler’s 3rd law:

P^2 = a^3

P^2 = (45.8)^3

P^2 = 96158.552

P = \sqrt{96158.552}

P \approx 310 \text{ years}

So, the orbital period of Makemake is approximately 310 years.