gen-define-kepler-s-laws-of-orbital-motion-and-give-one-short-wo-11ba

Define Kepler's laws of orbital motion and give one short worked example.

# Kepler's Laws of Orbital Motion ## The Three Laws **1. Law of Ellipses:** All planets move in elliptical orbits with the Sun at one focus. **2. Law of Equal Areas:** A line joining a planet to the Sun sweeps out equal areas in equal times (planets move faster when closer to the Sun). **3. Law of Periods:** The square of a planet's orbital period is proportional to the cube of its semi-major axis: $$T^2 \propto a^3$$ or more precisely: $$\frac{T^2}{a^3} = \frac{4\pi^2}{GM}$$ where T = orbital period, a = semi-major axis, G = gravitational constant, M = mass of central body --- ## Worked Example **Problem:** Mars has an orbital period of 1.88 Earth years. Using Kepler's Third Law, find Mars's average distance from the Sun (Earth's average distance = 1 AU). **Solution:** For any two planets orbiting the Sun: $$\frac{T_1^2}{a_1^3} = \frac{T_2^2}{a_2^3}$$ Let subscript 1 = Earth, subscript 2 = Mars: - $T_1 = 1$ year, $a_1 = 1$ AU - $T_2 = 1.88$ years, $a_2 = ?$ $$\frac{(1)^2}{(1)^3} = \frac{(1.88)^2}{a_2^3}$$ $$a_2^3 = (1.88)^2 = 3.53$$ $$a_2 = \sqrt[3]{3.53} = 1.52 \text{ AU}$$ **Answer:** Mars orbits at an average distance of 1.52 AU from the Sun.