1. Kepler’s 3rd Law Examples

2. 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 years. What is its average distance from the sun?
𝑇 2 = 𝑟 3 ;𝑇=𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑝𝑒𝑟𝑖𝑜𝑑, 𝑟=𝑚𝑒𝑎𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑢𝑛
= 𝑟 3
= 𝑟 3
𝑟=72.35𝐴𝑈

3. 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
= 4 𝜋 𝑥 10 −11 𝑘𝑔 −1 𝑚 3 𝑠 2 ×5.98𝑥 𝑘𝑔 𝑟 3
=9.898𝑥 10 −14 𝑟 3
7.54𝑥 = 𝑟 3
𝑟=∛7.54𝑥 ≈4.23𝑥 10 7 𝑚
𝑇=86400𝑠
𝐺=6.67𝑥 10 −11 𝑘𝑔 −1 𝑚 3 𝑠 2
𝑀 𝐸 =5.98𝑥 𝑘𝑔