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Kepler's Laws of Planetary Motion

Describe Kepler's laws and determine satellite periods mathematically

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Earth's Orbit

Earth’s Orbit

Credit: Laura Guerin
Source: CK-12 Foundation
License: CC BY-NC 3.0

While initially believed to have a circular orbit, the orbit of the Earth is actually elliptical. In 365 days, the Earth makes a complete orbit around the sun. During the Earth's one year period, the closet distance the Earth gets to the sun is 147 million km, while the greatest distance is 152 million km.

Amazing But True

  • According to Newton's law, every object in the universe attracts every other object with a force that is directly proportional to the square of the distance that separates them. Newton determined that the force between two bodies is given as:

\begin{align*}F = G \frac{M_1 M_2}{R_2}\end{align*}

where \begin{align*}G\end{align*} is the universal gravitational constants, \begin{align*}M_1\end{align*} and \begin{align*}M_2\end{align*} are the masses of the objects. Since the Earth is in orbit around the sun, we can define the force on the earth to be a centripetal force:

\begin{align*}\frac{Mv_2}{R_2} = G \frac{M_1 M_2}{R_2}\end{align*}

if we rewrite the speed of the orbit in terms of its period we get,

\begin{align*}T_2 = \left(\frac{4 \pi^2}{GM}\right) R_3\end{align*}

where \begin{align*}R\end{align*} is the radius for a circular orbit or for the length of a semi major axis for an elliptical orbit.

Show What You Know

Using the information provided above, answer the following questions.

  1. What force is keeping the Earth in orbit around the Sun?
  2. From the link provided, determine the reduced mass if \begin{align*}m_1 = 4 \ kg\end{align*} and \begin{align*}m_2 = 6 \ kg\end{align*}
  3. If the Earth had a uniform density, how would you expect the force of gravity to change as you get closer and closer to the center of the earth?

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Image Attributions

  1. [1]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0

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