Effect of Rotation on the Earth and Planetary Disk(s)

[Mazin Nasralla]

Hello,
This problem has probably been done before in different guises but I am struggling, and hoping for help. I am struggling to understand a phenomena which I am assuming is one and the same:

Equatorial Bulge of the Earth
Flat Plane in which the Planets rotate
Protoplanetary disks are flat
Disks around planets eg Saturn

I was reading something about the Tides, and the High Tides on Far and Near-side of the Earth and got that, but then started to think about the Earth's own equatorial bulge and started to look for similar phenomena that might explain it, but no I am still stuck.

In each of the examples we've got rotation. And I am constantly thinking about sedimentation in a centrifuge whereby the particles with the greater mass are "thrown" outwards. It does appear quite intuitive that if you rotate a deformable body it sill stretch out perpendicular to the axis of rotation like whirling pizza dough, but rather than keep looking for analogies, what's the physics behind this?

I've been trying to find out online and come across a few explanations which I semi-understand.

The Earth and the equatorial bulge is perhaps the easiest to have a go at, and it was my original problem. I have seen this argument and it does make some sense. I am repeating it to see if I can make more sense of it, and to see what other people think of it.

What's going on at the equator that's different to the poles? The rotational velocity is at its maximum here. Matter here is being accelerated through the max circumference in a constant time, and this acceleration is provided by the Earth's gravity And therefore the gravitational force at the equator is reduced by the centripetal acceleration meaning matter is less tightly held by the Earth's centre deforming the sphere. I know that the bulge itself increases the Earth radius, and therefore reduces g, but I want to know what caused the bulge in the first place.

Following this through, there must be a theoretical maximum rotation rate above which matter would just fly into space?

Is this explanation OK? Does it also cover the accretion disk, and centrifuge. I don't think it does. I looked online and got a bit dazzled by rotational frames of reference and fictional forces so I am hoping someone can help me understand. I've probably combined a few phenomena that are unrelated. As I was unable to understand one, I looked for something similar and failed to grasp the whole lot!

Hoping you guys can help. I've spent some time trying to figure this out and found an argument which makes some sense to me, but is it right? And does it explain the other phenomena?

 

[Bandersnatch]

Is this explanation OK?

It's not bad. And yes, there is a rotation rate that can break a body apart.

For an accretion disc it's pretty much the same. In fact, you can think of it as the stuff that's rotating too fast to coalesce into a compact body. It's the hypothetical body that 'was' torn apart by its rotation.

The centrifuge is different in that the forces are somewhat different - gravity pulls stuff down rather than towards the centre of mass of the material in the centrifuge, and there are obviously walls. But the principle of pulling away from the axis of rotation is the same as with planets and rotating discs.

 

[Mazin Nasralla]

Thanks for replying.

So in the accretion disk, the matter outside the plane of rotation experiences a higher gravitational force and is either pulled towards the disk or the central star leaving behind a disk, the region where the gravitational force is at a low?

 

[Bandersnatch]

No, that's not a correct way of seeing it. A body is always attracted in the direction (force is a vector) of the greatest (net) force.
If you think of the system in terms of centrifugal forces 'reducing' the gravitational force, they do so only in the plane of rotation (edit: by which I mean directions parallel to this plane, not the plane as a region in space), so what it means is that the rotation changes the direction a test particle 'thinks' the greatest force is, and tries to go there.Perhaps this simulation will help:
https://www.khanacademy.org/computer-programming/challenge-modeling-accretion-disks/1180451277
It's a program that simulates gravitational attraction between many particles in a slowly rotating cloud. Notice the behaviour. Remember that there is just one type of real force at work here - gravitation.

 

[Mazin Nasralla]

In the accretion plane, the gravitational force is at a low because of the centripetal acceleration, and therefore matter tends to accumulate in that plane?

The mass ends up where the force moves it so the conclusion must be that forces are directing the mass to this horizontal plane. Gravity itself is not affected by the rotation of the central mass (not according to Newton's equation anyway) so another force, or forces must come into play. That plane is the equivalent of the Earth's equatorial plane and marks the plane where the centripetal acceleration is at its maximum and using the Earth analogy the gravitational acceleration is therefore reduced in this zone and so matter accumulates here.

If I've got that wrong, can you let me know? Thank you

 

[Bandersnatch]

In the accretion plane, the gravitational force is at a low because of the centripetal acceleration, and therefore matter tends to accumulate in that plane?

No. I thought you might read it like that. See the edit in my post above. The plane of the disc is not a 'place with the lowest force'. There are no places with forces, there are directions. Matter accumulates in directions of greatest forces.

Gravity itself is not affected by the rotation of the central mass (not according to Newton's equation anyway) so another force, or forces must come into play.

Only in a rotating reference frame you get another (fictitious) force - the centrifugal force. If you use this frame to analyse the system, you can say that all particles in a rotating disc are not attracted to the axis of rotation (net force = 0), but are still attracted to the plane (force is greatest in that direction).

In an inertial, non-rotating reference frame (which means you're not rotating with the whole thing, but sit, stationary, outside), the only force is gravity.
The particles in this frame of reference have angular momentum around the axis of rotation that prevents motion in the radial direction. That is, they have some velocity that keeps them from falling towards the axis of rotation.
They don't have any such angular momentum in any other direction, so their motion is not constrained there.

Both descriptions are equivalent. Stick to the one that clicks better for you and try not to mix concepts between those two.

 

[Mazin Nasralla]

If the only force is gravity then why doesn't all matter accelerate in its direction?

I understand that the matter has angular momentum by virtue of its rotation, but could not the matter move inwards, and the whole system rotate faster conserving angular momentum. If the matter remains at average equidistant from the axis of rotation it must be balanced - as you say with a net force of 0, but I don't see what the counterpart is to the gravitational force. Sorry!

 

[Bandersnatch]

If the matter remains at average equidistant from the axis of rotation it must be balanced - as you say with a net force of 0

No! That is not a true statement. With force = 0 the velocity vector remains constant. Remember that most quantities in motion are vectors. Velocity has got a direction as well as magnitude. In a rotating motion in an inertial frame the velocity vector of test particle is non-zero and constantly changing due to the force pulling it inwards. If the force in the radial direction were 0, the particle would just fly off in a straight line.

Only in a rotating frame the you get a zero velocity vector, and you can say the force is zero because gravity cancels out with centripetal force.

If the only force is gravity then why doesn't all matter accelerate in its direction?

It does. But some matter has some velocity in some directions, like those rotating particles in direction tangential to the rotation. All the force is 'wasted' on curving the paths of those particles, and nothing is 'left' to bring them closer.
The problem with this way of putting it is that one might begin to think that the gravity is actually, really weaker in the radial direction. It's not. After all, it does all that curving of a straight path into a circle.

 

[Mazin Nasralla]

Thanks. I understood both statements in of themselves. Hopefully when I reconsider the whole phenomena it will fit together. I shall leave you in peace now!