Saturday, April 20, 2013

The Curious World of Artificial Gravity

Writing about space stations with artificial gravity, its fairly common to see people restrict themselves to a diameter of 500 meters or larger.  This follows mathematically from a commonly cited figure that Coriolis forces can be hard to tolerate at more than 2 rotations per minute.  If you require the production of 1 Earth gravity, then the required size is about 225 meter radius, and 450 meter diameter.  That is probably fairly reasonable.  It would be cheaper in terms of material to make stations even smaller, but there are a few reasons that very small habitats could be awkward, to say the least.  Let's look at the basic parameters of this size:
  • Radius set to exactly 250 meters
  • Edge velocity would be about 50 m/s, or 110 mph
  • The change in pressure from the center to the surface is about 1.6% of an Earth atmosphere

For the pressure change from the center to surface, it has to be assumed that the air rotates along with the cylinder.  This is yet another good reason to illustrate these tubes in the way I have in the introduction, that is, a tapered end so that the edge itself isn't directly exposed to the atmosphere.  If the ends are open the the air partially doesn't rotate with the tube, and my view is that these are hurricane-like conditions.  The hydrostatic head from a few 100 meters is a lot, which is effectively what would drive currents.

The really interesting metric is the edge velocity.  Imagine that you hop in a train that is moving in the direction opposite of rotation in this world.  If the train accelerates to 110 mph, it will become weightless.  It still has to work against the drag force from the air (air rotates with the tube).  This phenomenon would exist for any moving thing to some extent.  Your "weight" decreases the faster you move in the direction opposite of rotation, because after all, you're closer to being stationary than the rest of the tube.

Car or Plane Example

It's hard to imagine exactly exactly how this would play out, but that's what we have physics for!  For an electric car, the friction forces include aerodynamic drag, drag on the tires, and some internal losses.  The drag force is fairly predictable as proportional to the density of air times the velocity squared.  The force friction force from the tires, however, is roughly constant.  This isn't a huge surprise when you consider that the concept of coefficient of friction supposes a constant force irrelevant of speed.  But that concept also supposes the force is proportional to the normal force ("weight" in other words) - and this is where things can get interesting.  Let's look at the case of a car driving in the opposite direction of rotation.  In this situation, we can formalize a few things:
  • Weight of the car is equal to its acceleration, which is the speed of the rotating tube, minus its speed, squared times the radius.  R (V-v)^2
  • The drag from the wheels is proportional to the weight
  • The aerodynamic force is proportional to the square of velocity

I listed the constants for a typical car modeled with this perspective.  You can look at things in terms of force or power, but I prefer power because it reflect directly the amount of throttle you have to put into a car to keep it going at some given speed.

We also need to distinguish between a plane and a car, as well as address the issues with traction.  Driving a car is pushing it forward with the force of wheels on the ground.  This would stop working entirely as one approached 110 mph, the speed of the tube, because they would be near zero-gravity.  Tires only world with gravity.  The wheels losing grip and spinning isn't the only problem, you would also lose the ability to steer without slipping (and maybe tipping).  The opposite thing would happen as a car moving in the direction of rotation, the functional weight would increase, the grip of the wheels on the ground would increase, the weight the people feel in the car would increase, and you could steer tighter.  These cases are interesting to me, so I plotted them together here.

Power dissipation when moving against the direction of spin, versus with it
red - moving in direction of spin
blue - moving against direction of spin

A plane is different because it takes off while exerting force on the air.  The forward thrust is thus not dependent on the traction from the wheels.  Of course this is necessary because the plane plans to leave the ground.  You can notice in the above graph that the power required to drive is always increasing with increasing speed.  That's fairly "normal", and it's also what we would expect for a typical car driving in a 500 meter tube.  But to imagine a very curious case, I took the above car model and reduced the air friction by a factor of 10.  The effect this has is to make the tire friction dominate through more of the speed range.  But the tire friction decreases with increasing speed moving against the direction of ground motion.

Power dissipation when moving against the direction of spin
using tire friction + air friction, more tire friction than last graph

If you imagine that this model fits a plane, you discover that something truly strange happens.  At a certain speed you obtain a local maximum in power.  If you were accelerating, slowly increasing the throttle up to that point, then once you reached the point you would experience increasing speed with no increase in throttle.  By the time the system equalized, you would nearly be flying.  This brings up another good point, in an artificial gravity tube you can have wingless flight.  In fact, most flying things we're familiar with can hit 110 mph.  Basically, in this world, planes wouldn't have wings.  It's disputable whether it would have any "cars" in the first place.

Cross-Habitat Structures

You need to consider that for any artificial gravity environment, it will likely make sense to have structures that span the diameter of it.  There are a few reasons why, and the structural motivation is compelling.  Imagine that the artificial gravity tube was built with a uniform strength, designed for a given mass-thickness of load.  If you wanted to expand operations in a location further, it could be difficult.  In fact, putting any large amount of matter in a singular place would cause strange kinds of sagging in the structure.  The mass distribution really has to be uniform along the circumference, but structures that span the diameter could be the exception to that.  By building a tall building on both sides, they could be each others counterweight.  Considering that these would be held by tensile (and not compressive) strength, they could actually be cheaper to build than their equivalent building on Earth.

There is an obvious limitation to this, which is that the gravity decreases as you go higher.  If you are co-moving with the structure, then gravity changes linearly with radius.  So if you move up half the radius to the center of the tube, the gravity will decrease by half.  A 500 meter diameter tube, thus, could host a building 500 meters "tall" but the floors would have gravity evenly distributed along the spectrum of zero gravity to full gravity.  On Earth, we have a handful of occupied buildings that rise over 500 meters at the top, but the rest of mankind's buildings are shorter than that.  Examples:
  • Burj Khalifa                       829.8 m
  • Mecca hotel                      601 m
  • One World Trade Center   541 m
  • Petronas Twin Towers       458 m

If you imagine that gravity as low as 0.9g is tolerable, then you can only have 50 meters of usable floors in a cross-habitat structure, and only 25 meters on each end.  Most typical city centers with a decent skyline have a few buildings higher than 100 meters.  So in short, if an artificial gravity tube was to have the economic density of something like Manhattan, it's likely necessary to have a diameter larger than 500 meters, just to fit the buildings in.

Others have written about staircases that transition to zero gravity.  This idea actually occurs in several sci-fi ideas as well as serious discussions about space habitats.  As for a 500 meter diameter habitat, climbing ropes typically have lengths 30 to 80 meters.  While 500 meters is a lot, it's not unthinkable.  It's also fun to imagine how you could string such things ad-hock.  Anchor one end, then walk around the circumference holding the rest of the rope.  With those tied in sufficiently, you could climb straight across the habitat right away.

Spooky Vertical Motion from False Fields

A good read on the funny behavior of things in artificial gravity can be found here.  One of the strangest thing about living in an artificial gravity environment would be that when you drop something, it deflects some amount.  This reference illustrated this for two cases with 1g of gravity.  The dots show the location every 1/10th of a second.  It illustrates something falling from head-level, and the movement of a body's center of mass when jumping up straight.

Movement of drop from head-level and feet movement with jump
both are for 1g of gravity, taken from

There are a number of ways that you could go about establishing the physics of such a place.  Logically, you might just want to look at some object, falling or moving, and then consider how the ground would move relative to it.  Another way is a full transformation of reference frames, which is a daunting task in many ways.  The ground both rotates and accelerates.  This nature is shared with orbiting reference frames that are tidally locked.  I tried my best at the problem on Physics SE.  It was surprising how difficult it is to do the full transformation of reference frames.  I want to keep this simple, so I solved a simplified case, to exclusively answer the question of how much an object dropped from head-level would deflect.

Diagram and calculations for deflection of dropped object

Walking through the math, you can observe that the dropped object follows the tangent line from the point where it was dropped.  Until it hits the floor it maintains its original velocity.  The distances that the object and the ground (the point that starts out directly below the object) travel before the hit are illustrated in red.  You can qualitatively observe that the dropped object travels the same total distance that it would have otherwise had it been held at its original elevation.  That allows the establishment of the needed relationships with simple trigonometry relationships.  These are more-or-less summarized in that image.  Then using these, as well as some other substitutions from the fact that the tube has a known gravity, I put the expression in two different forms.

Expression for the deflection distance for an object dropped at delta above the ground
Second expression is the deflection in terms of a fraction of the original height

Plugging in the numbers, this comes out to be quite significant.  If you use a drop height of 1.5 meters (about the shoulders of a human), here are the deflection values for a tube rotating at various rpm values, in order to get correspondence with my reference case and the above 1 and 4 rpm examples.

Given: one Earth gravity, 1.5 meter release height
  • 4 rpm, 50 m radius, 25 cm deflection or 16%
  • 2 rpm, 225 m radius, 11 cm deflection or 7.6%
  • 1 rpm, 900 m radius, 6 cm deflection or 3.8 %
An important observation here is diminishing marginal returns.  The size you have to build the structure goes up very fast just to slow the angular speed down a little.  Plus, the material requirements grow proportionally to radius.  That means that for increasing the material requirements by about a factor of 4 you only halve the deflection effect.  Clearly this would be difficult to do away with entirely.  I think it suffices to say that people living there would always perceive the direction of motion.

The two examples discussed here of a car and a falling object are useful mathematically.  The reason being that this part of the fictitious forces comes from a cross-product of the object's velocity with the tube's angular momentum.  The angular momentum goes in the direction of the tube's axis.  That implies that you don't experience this type of force when moving in the direction of the tube's axis, and this is correct.  If you were to throw a ball, there is "distortion" in its path if thrown up or in a tangential direction, but not really when moving down the length of the tube.

Maybe this isn't all that bad of a thing.  I am reminded of a particular experiment which attached a special belt to study participants.  The belt was lined with vibrators.  Only the vibrator on the north side would vibrate, but it would do this all of the time.  It was found that the participants adjusted to life with the belt to an extent that was downright creepy.  Their brain had found a way to use the new sensation to help orient themselves and navigate as a classic example of sensory substitution.  I think the most amazing detail is that they reported nausea after they took the belt off.  It's not too far of a jump to imagine that this same thing may happen to people living in an artificial gravity environment.  Their minds would constantly be searching for the direction of rotation.  In fact, it may be stranger for them to consider our world than for us to consider their world.  On Earth, we seem to have nothing to distinguish between directions on the horizon other than a weak magnetic field from the center of Earth.  In artificial gravity you have a natural compass!

These kind of questions led others to propose the idea of Planetary chauvinism.  This is mostly proposed because of the difficulty of climbing in and out of gravity wells.  But it's worth mentioning that the "weirdness" of artificial gravity probably disappears depending on the amount of time you spend there.  We only speak about building things similar to Earth because Earth is all we know.  We can't judge a strange new world without ever have lived in it.  Being weird may be a good thing.  That leads me to the next topic - a mixed gravity environment, which I hope to write about in a future energy.  The gravitational balloon would be a truly mixed environment where one could go from gravity to no gravity and back again... even multiple times during a workday.  People would not only be perfectly comfortable in artificial gravity, but with zero gravity as well, not to mention everything in-between.

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