A fictitious force in an orbiting frame of reference that is not rotating

Question

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.

Answer 1

I think in this situation the name of the fictious force is not "centrifugal". But this is not really important: your reasoning and calculations are correct.

I think so because I checked the definition of "centrifugal force", and the definition I found mentions "rotating frame of reference". In this case the frame of reference is not rotating, it just moves with acceleration which depends on time. That's why I think "centrifugal" name should not be used in this case.

But if someone else would find another slightly different definition and claim that the force is centrifugal I would not argue!

One more note. You write:

Also, this fictitious force cancels out the centripetal acceleration of the planet.

I understand what you mean, but I think this statement is not correct. The force can cancel out some other force, but not acceleration. I would put it in other words: in our not inertial frame of reference the fictional force cancels out the gravitational force of the star applied to the planet. Sum of the forces is zero, so the planet is not accelerating in our frame of reference.

Answer 2

I concur with the earlier answer by lesnik that the frame of reference you are describing is not a 'rotating frame of reference', even though superficially it may seem so.

As lesnik points out, what you describe is a frame of reference that is undergoing linear acceleration. Sure, over time that acceleration changes direction, in a cyclic way, but it's still linear acceleration. There is no special name for that, only the straightforward description 'linear acceleration'.

General discussion of laws of motion:
We have that inertia is the prime organizing principle for our understanding of motion.

Every theory of motion we have has the following in common: motion is described relative to an inertial frame of reference. Any attempt at an equation of motion that doesn't reference an inertial frame isn't a valid equation of motion.

This can be seen as follows: whenever an equation of motion is used that deals with motion relative to some rotating frame of reference the equation contains the angular velocity $\omega$ of the rotating frame relative to the inertial frame of reference.

In the end the inertial frame of reference is always the reference of motion. Either directly, or in the form of the inertial frame being the reference for the angular velocity of the rotating frame.

Causality

The distinction between inertial frames of reference and accelerating frames of reference has implications for what we recognize as causing motion.

As we know, the expression 'fictitious force' refers to the extra terms in an equation of motion for motion relative to a frame of reference that is rotating/accelerating.

That wich causes acceleration with respect to an inertial frame of reference counts as a force. On the other hand, a fictitious force does not cause acceleration. A fictitious force is a calculational device; in some situtations it can simplify a calculation. Another way of saying this: a fictitious force does not play a role in causality.

When Newton presented his description of gravity he was effectively arguing that gravity should be regarded as a force. The Earth is circumnavigating the Sun, due to the gravitational attraction from the Sun. But you don't actually feel the gravitational attraction from the Sun, why is that? Newton pointed out that that is explained entirely if we recognize that inertial mass is equivalent to gravitational mass. If those two are equivalent then your own tiny body and the entire body of the Earth will accelerate towards the Sun with exactly the same acceleration. You are not accelerated relative to the Earth, that is why you don't directly feel the gravitational attraction from the Sun. (The Earth as a whole does feel a secondary gravitational effect, resulting in tidal motion.)

Newton argued that if the laws of motion hold good then the common center of mass of all the masses in the solar system is in inertial motion. Notice that Newton did not claim that the center of mass of the Sun by itself is in inertial motion. When two or more masses are orbiting each other they are all being accelerated. It is the common center of mass of all of the mass in the system that is in inertial motion.

So this is the newtonian point of view:
Gravity must be classified as a force, because we infer that gravity causes acceleration with respect to an inertial frame of reference. At the same time, during free fall you don't feel gravity since gravitational mass is equivalent to inertial mass.