Reply to FyzixFighter 

FyzixFighter, you are making some of the same mistakes as Brews. You seem to think that we can take that four part expansion of acceleration in terms of polar coordinates and read some physics into it. We can't. The polar terms tell us nothing about any particular motion.

But they are nevertheless very useful. We can use the expressions in a properly modelled differential equation that involves real physical input. The radial convective term is used as a centrifugal force in the two body gravity problem.

You seem to think that because it is derived with a negative sign that it has to mean centripetal force. That is not so. The negative sign is a consequence of the outward positive convention for the position vector.

The expression is merely the mathematical form for a force that changes the direction of motion and acts in the radial direction.

When we construct the planetary orbital equation, we end up with a radial gravity force inwards and a centrifugal force radially outwards. We have a scalar equation in the radial length and the two forces equate to the second time derivative of the radial length. You have got too much bogged down in trivia over matters like whether or not the second time derivative of radial length is entitled to be termed radial acceleration or not.

It doesn't matter. Just concentrate on the equation and solve it. And in doing so, you will see that inward radial gravity is working in opposition to an ongoing outward radial centrifugal force.