# Why do planets have a greater linear velocity closer to the Sun?

I get that the planets need to have a higher velocity to escape the gravitational well which is deeper when closer to the sun. What I don't understand is what causes this higher velocity. Am I missing something simple?

When the planet moves from aphelion to perihelion it is accelerated by the attraction of the sun. The attraction force has a tangential component in the same direction as the velocity. You can think that the planet falls towards the sun. When it moves away from the sun the tangential component of the force is slowing down the motion, it is opposite to velocity. It is like trowing up an object, if slows down as it goes higher.

Earth goes around sun in an eliptic orbit, with sun being one of the foci of that ellipse. Now, maximum distance of earth from sun is called Apphelion distance, and minimum distance of earth from sun is Perihelion distance.

Let's take earth and sun as one system, and ignore external gravitational forces from other celestial bodies, since they are negligible.

So effectively : $$\sum\Gamma_{\text{external}}=0$$ where, $$\Gamma$$ is torque applied by other celestial bodies.

So we can write: $$\sum L=\text{constant}$$ where $$L$$ is total angular momentum of our system.

Let, earth's linear velocity at perihelion be $$v_1$$, and distance be $$r_2$$, and similalry for velocity at apphelion be $$v_2$$, and distance be %r_2\$

From second equation:

$$m_ev_1r=m_ev_2r_2=\text{external}$$

Now by simple alegbrae, we can see that $$r_2=r_{max}$$, so obviously, $$v_2=v_{min}$$ for our product to stay constant.

Similarly, $$r_1=r_{min}$$, and $$v_1=v_{max}$$.

Or generally:

$$r \propto v^{-1}$$.

Hope it helped you out.