### Earth?s Orbit

Initially believed to have a circular orbit, the Earth's orbit is actually elliptical. As the Earth orbits around the Sun in 365 days, its distance from the Sun varies between 147 and 152 million kilometers.

#### 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:

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

\begin{align*}G\end{align*} is the universal gravitational constants and \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*}

\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.

- Learn more about the Earth's elliptical orbit: http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html

#### Show What You Know

Using the information provided above, answer the following questions.

- What force is keeping the Earth in orbit around the Sun?
- 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*}.
- 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?