Say we have an inertial frame of reference where there's a motionless star at the center and a planet is in circular orbit around it. We further assume that the planet is not rotating in the inertial frame, it is just orbiting. E.g., if this planet is a cube aligned with the inertial frame's major axes, it remains aligned with them at all times.
The position of the planet at time $t$ is given by the vector $r(cos(wt),sin(wt),0)$, where $r$ is the radius of the circular orbit, and $w$ the angular velocity.
My question is related to the fictitious forces that are present in the non-inertial frame of an observer on this planet - a non-inertial frame that is orbiting, but not rotating.
Here's how I computed it: Say some object's position is described by a function $p(t)$ in the inertial frame. Switching to the planet's frame of reference, we get the position $p(t)-r(cos(wt),sin(wt),0)$.
We can obtain the object's acceleration by a second derivation of both terms: $a(t)+rw^2(cos(wt),sin(wt),0)$, where $a(t)$ is its acceleration in the inertial frame. Therefore, the second term, $rw^2(cos(wt),sin(wt),0)$, is the fictitious acceleration caused by a fictitious force. It is opposite the centripetal acceleration of the planet.
And my question: is this fictitious force a "centrifugal force"?
Note that it is very unlike the textbook centrifugal force. The fictitious acceleration $rw^2(cos(wt),sin(wt),0)$ applies equally to all objects in the scene, regardless of their position: the same magnitude and same direction everywhere (for a given $t$). This is unlike the centrifugal force that does depend on the object's position. It is always directed away from the center, and proportional to the distance from it.
If it's not centrifugal, then does it have a name?
Also, this fictitious force cancels out the centripetal acceleration of the planet. Isn't it a simple explanation as to why being on an object in orbit 'feels' like a force-free inertial frame in Newtonian mechanics (I know the explanation of general relativity is different)? I haven't seen this explanation anywhere.