The Movement of Planets

By applying the Newton's 2nd law to a planet considered

F S / P ⃗ = m a ⃗ vec F _{S/P}=m vec a

The force being radial

G m M S r 2 = m a n → a n = G M S r 2 G {m M_{S}} over { r^{2}}=m a_{n} rightarrow a_{n}=G { M_{S}} over { r^{2}}

The acceleration of the planet in its movement is only radial, directed towards the center of the sun.

Hence, the movement of the planets around the sun can be considered a uniform circular motion

Also The speed and period of a planet is defined as follow

v = GM S r , T = 2 π r v v= sqrt{ { GM_{S}} over {r} } , T= {2 %pi r} over {v}

T 2 r 3 = 4 π 2 G M S = cst { T^{2}} over { r^{3}}= {4 %pi ^{2}} over {G M_{S}}= cst

Note

From this equation T 2 r 3 = 4 π 2 G M S = cst { T^{2}} over { r^{3}}= {4 %pi ^{2}} over {G M_{S}}= cst , we can conclude that Kepler's 3rd law is verified