gen-define-kepler-s-three-laws-and-give-one-short-worked-example-cfa5

Define Kepler's three laws and give one short worked example.

# Kepler's Three Laws of Planetary Motion ## The Three Laws **1. Law of Orbits (First Law)** Planets move in elliptical orbits with the Sun at one focus. **2. Law of Areas (Second Law)** A line connecting a planet to the Sun sweeps out equal areas in equal time intervals. **3. Law of Periods (Third Law)** The square of a planet's orbital period is proportional to the cube of its semi-major axis: $$T^2 = \frac{4\pi^2}{GM}a^3$$ Or comparing two planets: $\frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3}$ ## Worked Example **Problem:** Mars has an orbital period of 1.88 Earth years. What is its average distance from the Sun? (Earth's distance = 1 AU) **Solution:** Using the Third Law comparing Mars and Earth: $$\frac{T_{Mars}^2}{T_{Earth}^2} = \frac{a_{Mars}^3}{a_{Earth}^3}$$ $$\frac{(1.88)^2}{(1)^2} = \frac{a_{Mars}^3}{(1\text{ AU})^3}$$ $$3.53 = a_{Mars}^3$$ $$a_{Mars} = \sqrt[3]{3.53} = 1.52 \text{ AU}$$ **Answer:** Mars orbits at an average distance of 1.52 AU from the Sun.