Kepler’s 3rd Law Examples

𝑇 2 = 𝑟 3 ;𝑇=𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑝𝑒𝑟𝑖𝑜𝑑, 𝑟=𝑚𝑒𝑎𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑢𝑛 Example 1 A dwarf planet discovered out beyond the orbit of Pluto is known to have an orbital period of 619.36 years. What is its average distance from the sun? 𝑇 2 = 𝑟 3 ;𝑇=𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑝𝑒𝑟𝑖𝑜𝑑, 𝑟=𝑚𝑒𝑎𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑢𝑛 619.36 2 = 𝑟 3 383606.8= 𝑟 3 𝑟=72.35𝐴𝑈

Example 2 From a telecommunications point of view, it’s advantageous for satellites to remain at the same location relative to a location on Earth. This can only occur if the satellite’s orbital period is the same as Earth’s period of rotation, approximately 24 hours. At what distance from the center of can geosynchronous orbit be found? 𝑇 2 = 4 𝜋 2 𝐺 𝑀 𝐸 𝑟 3 86400 2 = 4 𝜋 2 6.67𝑥 10 −11 𝑘𝑔 −1 𝑚 3 𝑠 2 ×5.98𝑥 10 24 𝑘𝑔 𝑟 3 7464960000=9.898𝑥 10 −14 𝑟 3 7.54𝑥 10 22 = 𝑟 3 𝑟=∛7.54𝑥 10 22 ≈4.23𝑥 10 7 𝑚 𝑇=86400𝑠 𝐺=6.67𝑥 10 −11 𝑘𝑔 −1 𝑚 3 𝑠 2 𝑀 𝐸 =5.98𝑥 10 24 𝑘𝑔