Centrifugal Field Divergence for a Rotating Planet

Centrifugal Potential is a false potential within the context of a rotating reference frame, which can be thought of as coming from the false Centrifugal Acceleration.  I was reworking through some of the details of this and was going to post a question to Physics Stack Exchange, but it turned out that my knowledge was complete enough to not really merit a question, because I had nothing to ask, so I'll post the work here.

This has application to a rotating planet.  The mathematics for Centrifugal Potential is general to any rotating field.  I'll call it u_r.  Call the angular speed omega, and distance from the axis of rotation rho. The potential can be arrived at several ways, and they all give the following.

I would like to find the Laplacian of this.  This would be the same thing as the divergence of the Centrifugal Acceleration.  With the potential in the above form, if I understand correctly, we have to use the cylindrical version of the Laplacian.  The cords (rho,phi,z) are (distance from axis, azimuth angle, and vertical position).

The last 2 derivatives are zero, and we can expand the first.

I questioned the factor of two, but now I think that's probably correct.  This would be useful to compare this to the divergence of Newtonian gravity which can be found from Gauss' law, which I'll call u_g.

If we use Gauss' law again, then would we be able to make a 100% true statement about the integral of gravity over the surface of a rotating planet?  We would need to take dot product of gravity with the surface, but if it fits the hydrostatic condition then all gravity is normal anyway.

The two volumetric terms only depend on density.

Is it then permissible to write the following?

With this, we have two extremely interesting terms.  I think the most interesting thing is the ratio of them.  For instance, with Saturn, I calculate the rotational term is about 10% that of the gravitational term.  This means that the rotation subtracts 10% of the surface gravitational flux that its gravity creates.

It's hard to extend that statement very much further.  We would be tempted to say that someone standing on the surface of Saturn (ignoring the actual complications in doing this) would weigh 10% less due to the planet's rotation, but that's not completely correct because gravity isn't the same everywhere to begin with.

This does have some application to gravity balloons.  We can apply the concept of Gauss' Law balance to different volumes.  For instance, if we assume that the inner pressurized volume has effectively zero mass, then that implies that a rotating gravitational balloon would have a net outward field on the walls.  That means that over time things would avoid the center and move toward the walls.  That's an interesting effect and it would have a positive impact on stability, causing the Raleigh-Taylor effect to no longer want to mix the walls up.

If you wanted to take this to the absolute extreme case, you could imagine creating artificial gravity on the inside walls by rotating the structure fast enough, while still holding it together through self-gravitation.  In principle, this could possibly work, but to make Earth-like gravity, the mass required would be much greater than Earth itself.

Clover Orbits and Heat Removal

Without engineered solutions, any gravity balloon is thermally isolated.  The 10 km of rock would prevent any significant heat loss by conduction, and there is no other mechanism to get the heat out.  An engineered solution would have to entail some kind of matter exchange.  For small conceptions of the idea, this could probably be done in batch processes, which would also make more sense in the context of access tunnels and the airlock structure.  Once it gets much larger, however, there's a bit of a different challenge.  As the size grows, the habitable volume grows with the cube of the linear dimension.  In light of that, I want to talk about solutions for cooling that scale very well with large gravity balloons.

Pipelines in Zero Gravity

The idea of machines (and industrial society in general) spanning large swaths of volume in zero gravity is an exotic idea.  You can reliably have a process that involves huge exchanges of mass and even huge velocities while at the same time putting very little energy into it, and actually having very little rigidity to it.  The engineered systems in this environment could be large, complex, but fragile.  Imagine a normal train and the sheer impossibility of stopping such a thing, and multiply that concept by a thousand.  People won't even be worried about such huge huge things moving about because it would be so easy to move around it.  This is a true embodiment of the relative-ness of motion.  For a simple example, image a large pipeline of things moving in a giant circle.  This pipeline could be a simple ring, but with a thickness much smaller than its radius.

Moving from the inner surface to the outside surface requires doing work because there is a gravitational field.  If you're moving individual objects it also requires the energy to move in and out of the airlock, but we can imagine a smooth pipe that doesn't have this burden.  You don't necessarily have to put in the work against the field every time.  If the stuff you're moving is already moving sufficiently fast then it can make it out without much problem.  In a simplistic sense, we can imagine a set of tunnels in the rock, some going out and some going in.  After the cooling mass goes out a hole, then it makes its way to the "in" hole moving along the simple geodesic.  In this case, by geodesic, I mean a path traveled in free-fall.

Pragmatically, making a large pipeline of material going through an access tunnel would be quite an engineering feet, but mostly due to the challenge of keeping the air in.  In fact, the challenging of maintaining a pressure boundary over a moving surface is quite common in engineering and there is certainly no simple solution.  However, in the case of the shell of a gravity balloon, there would be abundant space over which to implement this solution.  Ultimately, this could provide an easier way for even people to get from the inside to the outside than by constantly messing with batch-process airlocks.  Another complication is that the shape would have to deform as the speed changes due to gravity.

Orbits Moving In and Out

Interested in how you would actually have to configure these holes, I asked about the problem in a limit case on Physics Stack Exchange.  To make the answer simple enough, we imagine that the rock is thin relative to the overall dimensions of the thing.  That now fits the same problem of a charged particle orbiting in and out of a spherical wire mesh at some voltage.  There was a very intriguing answer to that question.  Basically, there are huge number of ways to configure these odd orbits.  Not only can you have any given number of "dips" outside the shell, but for a given periodic number, you can have any degree of roundness you like.

The most practical case for industry in a gravitational balloon is obvious to me as the 2-exit orbit.  I've dubbed this a "2-gon" in the question on Stack Exchange because it exists within a family of chopped polygons.  This would be an ideal orbit for a pipeline into and out of a gravity balloon because it hits the walls at a high angle, limiting the distance of tunnels needed, and it can have a long path length within the center.  You can see in the question that for a larger number of dips it skirts the edge more, as opposed to making deep secant lines.

The 2-gon orbit image by Physics SE user JJ Fleck  

For gravity balloons of more moderate sizes, a large fraction of the orbit would be spent within the rock itself.  I tried to give some thought to the orbital shape within that rock.  I asked a question about a similar case of something orbiting within a field that was proportional in strength to the radius.  However, I realized that this doesn't fit the shell volume that I'm speaking of here.  The field is linear with radius but it is not directly proportional to radius.  That makes the solution for the orbit rather complicated.  Ultimately, it's sure to be a relatively small revision on what kind of shapes you can expect, but it needed to be mentioned.

The question remains of how practical this would be.  Particularly, how fast would such a cooling pipeline move?  We could use orbital dynamics, but there's another way to cheat and get an answer simply.  In the large case we know exactly what the gravitational field on the surface will be, provided we have the required internal pressure.  In this case the rock density doesn't even matter due to interesting mathematical conclusions.  The field is just:

$$g = 2 \sqrt{ \left( 2 \pi G P \right) } = 0.013 \frac{m}{s^2}$$

With this, I can generally answer the question of the orbit time for a coolant loop.  Again, for the large limit, there is a helpful simplification.  Just imagine the field on the surface to be constant.  Then we can get a time for the turn-around as well as a velocity.  The velocity should probably be kept well below the sonic velocity of the atmosphere, and ideally we would like to see round trip time less than a day or so.  To make this workable I assumed a generic velocity of 50 miles per hour, or 22.3 meters per second.  With that, the height a pipeline will protruce above the surface before naturally falling back down is calculated.

$$1/2 a t^2 = h \\ v = a t = sqrt( 2 h a ) $$

Numerically, plugging in the 50 mph figure:

$$ h = 1/2 v^2 / a \approx 20 km \\ h \approx \frac{1}{2} \frac{\left( 22.3 \frac{m}{s} \right)^2 }{ 0.013 \frac{m}{s^2} } \approx  20 km $$

This is a nice answer because it is on the order of the shell thickness in the first place.  That means that even if the case we're looking at isn't all that big, if the center bubble is just a few times greater than the shell thickness, the idea will be perfectly workable.  Now, with this speed we can imagine the travel times for certain things.  Even for a giant diameter of 250 km, it would still only take 3.1 hours to traverse the diameter at the speed.  You can then imagine multiplying this by 2 for the reverse trip.  Then, for the times spent outside the shell we'd be looking at about half the average velocity.  So credit about 10 km shell thickness, 20 km turn around distance, then a full orbit takes about 30 km of that distance.  This adds about .3 hours.  So for a full pipeline orbit through a 250 km gravity balloon, we'd be looking at a time on the order of about 7 hours.

There are several variables I haven't delved into, like the overall cross sectional size of such a space pipeline, but even with a normal "train" size, such a thing could be used to transport massive amounts of heat.  There are a lot of remaining complications, but it would scale very well.