ExampleCalculatingOrbitalPeriods 2025-03-01 06:19:43 2025-03-01 06:20:28 Example Kepler's third law of planetary motion % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \subsection{Calculating Periods} Imagine an object is traveling around the Sun. What would be the orbital period of the object if its orbit has a semimajor axis of $50$ AU? \subsection{Solution} From Kepler's third law of planetary motion, we know that (when we use units of years and AU) $$ P^2 = a^3 $$ If the object's orbit has a semimajor axis of $50$ AU ($a = 50$), we can cube $50$ and then take the square root of the result to get $P$ $$ P = \sqrt{a^3} $$ $$ P = \sqrt{50 \times 50 \times} = \sqrt{125,000} = 353.6 \,\, years $$ Therefore, the orbital period of the object is about $350$ years. This would place our hypothetical object beyond the orbit of Pluto. \subsection{Check Your Learning} What would be the orbital period of an asteroid (a rocky chunk between Mars and Jupiter) with a semimajor axis of $3$ AU? \subsection{Answer} $$ P = \sqrt{3 \times 3 \times 3 } = \sqrt{27} = 5.2 \,\, years $$ This article is a derivative work of the creative commons share alike with attribution in [1]. \begin{thebibliography}{9} [1] Fraknoi, Andrew, David Morrison, and Sidney Wolff. The Sky Above. In Astronomy 2e. Houston, Texas : OpenStax, 2022. \PMlinkexternal{The Sky Above}{https://openstax.org/books/astronomy-2e/pages/2-1-the-sky-above} \\ \end{thebibliography}