Question

speed at which any planet moves through space is constantly changing. In a perfectly circular orbit, the orbital radius of the planet would be constant and therefore so would be its observed angular velocity. In elliptical orbits, the angular velocity varies. In elliptical orbits, the orbital radius of the satellite will vary and therefore so will its angular velocity. The planet travels "faster" (greater angular velocity) when closer to the Sun, then "slower" (less angular velocity) at a more distant radius. According to Newton's 2nd Law, what is the underlying force behind this change in velocity?

Answer 1

The underlying force behind this change in velocity is the gravitational attraction between the Sun and the planet. In fact, the magnitude of the gravitational force is
$G \frac{Mm}{r^2}$
where 
G is the gravitational constant
M is the mass of the Sun
m is the mass of the planet
r is the distance of the planet from the Sun

This force provides the centripetal force that keeps the planet in circular motion, so we can write:
$G \frac{Mm}{r^2} =m \omega^2 r$
where the term on the right is the centripetal force, and $\omega$ is the angular speed of the planet. Simplifying the equation, we get
$G \frac{M}{r^3}= \omega$
And from this equation we immediately see that, when the distance of the planet from the Sun (r) is small, the angular speed $\omega$ is larger and the planet travels faster; vice-versa, when the distance is large, the angular speed is smaller, and the planet travels slower.

Answer 2

The underlying force behind this change in velocity is the gravitational attraction between the Sun and the planet. In fact, the magnitude of the gravitational force is

where 

G is the gravitational constant

M is the mass of the Sun

m is the mass of the planet

r is the distance of the planet from the Sun

This force provides the centripetal force that keeps the planet in circular motion, so we can write:

where the term on the right is the centripetal force, and  is the angular speed of the planet. Simplifying the equation, we get

And from this equation we immediately see that, when the distance of the planet from the Sun (r) is small, the angular speed  is larger and the planet travels faster; vice-versa, when the distance is large, the angular speed is smaller, and the planet travels slower.

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