### Author Topic: Artificial gravity habitat on a planetary body  (Read 14879 times)

#### punder

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##### Artificial gravity habitat on a planetary body
« on: 03/09/2015 07:00 PM »
MORE artificial gravity!  WooHoo!!      Apologies if this has been done before on this site or elsewhere.  Please send me the link if so.

I did some math to come up with the "floor shape" for a spinning AG habitat sitting on a planetary surface.  By floor shape, I mean the curve that enables someone to stand orthogonal to the surface and not feel any (significant) tugs to the side.

(It's not a hard problem--would make a good homework question for a calculus-based freshman physics class--but it was difficult for me, anyway.  It was also a lot of fun exploding the heads of myself and a few friends as we whiteboarded it out.)

The curve is a parabola, y = (w2/2Ag)r2 where:

w = angular velocity of the habitat
Ag = the planet's gravitational acceleration
r = distance from the hab's center of rotation.

You might ask, why the heck would you even do that?  Well, it might turn out that lunar gravity, for instance, isn't enough to support normal bone and muscle health.  Or, adults might be fine, but children--either gestating or growing up--might need more gee to develop correctly.  So there might be a reason to do it, or there might not.  Either way, the math and physics are interesting.

I’m doing a single-variable integration, setting w and Ag as constants.  Someone more edumacated might come up with a more general solution in which these are variables.

Here goes:

At any point on the floor, there are two vectors: Ag and centripetal acceleration Ac.
Ac = rw2.
The normal (to the floor) acceleration An is the sum of Ac and Ag.  This is the “target” acceleration, most likely 9.81m/s2.  Its slope is Ag/Ac.
The slope of the “floor curve” is perpendicular to this vector, so it is Ac/Ag = rw2/Ag.
Since you now have a slope, you can integrate to get the curve.  The integral of y’ = rw2/Ag is y = (w2/2Ag)r2.

That's it!  Now you can write an equation for any combination of angular velocity and local gravity to design “floors” for rotating habitats on the Moon, Mars, or any other planetary body.

For example, say you’d like a 1rpm habitat on the Moon, with a max “gravity” equal to Earth’s, 9.81m/s2.

Ac = sqt(9.81m/s2 - 1.63m/s2) = 9.67m/s2
r = Ac/w2 = 9.67m/s2/(2pi/60s)2 = 882m
y = (w2/2Ag)r2 = ((pi/30)2/1.63)r2
y = 0.003365r2
Subbing in, you get y = 2618m.

Now say your rotating pressure vessel is a torus, and the cross sectional diameter is such that the “bottom” point on the floor (min r) is 20m below the “top” (max r).

rmin = sqt(2598/.003365) = 879m
Acmin = (879m)((pi/30)2) = 9.64m/s2
Anmin = sqt((9.64m/s2)2 + (1.63m/s2)2) = 9.78m/s2

Again: if you see a horrible math or physics mistake (or a sup/sub problem or typo amounting to the same thing), please let me know.  Don’t be shy.      A diagram is attached to help visualize it.
« Last Edit: 03/09/2015 07:24 PM by punder »

#### KelvinZero

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##### Re: Artificial gravity habitat on a planetary body
« Reply #1 on: 03/09/2015 07:46 PM »
Thats quite cool. You could have an entrance at the bottom and just walk up to higher gravities.

This could almost be thrown in the "Manned Missions to Ceres" thread. Ceres has 3% gravity.

One issue, with a structure as large as I imagine this you might only be able to put it exactly at the planets pole to be aligned with the planets rotation or else have to deal with continual twisting forces in the bearings. Ceres rotates every 10 hours for example.

..or would it be possible to engineer the axis to precess like a wobbly spinning top exactly in line with the planet's rotation?

#### punder

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##### Re: Artificial gravity habitat on a planetary body
« Reply #2 on: 03/09/2015 08:12 PM »
Thats quite cool.

Thanks!

Quote
You could have an entrance at the bottom and just walk up to higher gravities.

This could almost be thrown in the "Manned Missions to Ceres" thread. Ceres has 3% gravity.

One issue, with a structure as large as I imagine this you might only be able to put it exactly at the planets pole to be aligned with the planets rotation or else have to deal with continual twisting forces in the bearings. Ceres rotates every 10 hours for example.

..or would it be possible to engineer the axis to precess like a wobbly spinning top exactly in line with the planet's rotation?

I did the numbers for the Moon and Mars at 1 and 2 rpm, and thought about Ceres (difficult not to think about Ceres these days) but didn't work that one out yet.

I don't have the chops to figure it out, but I imagine there would be negligible problems on a very large, slowly rotating body like the Moon.  On a small, low-g, fast-rotating body like Ceres, other forces might become a lot more important.

Edit: If the rim (or maybe even the hub?) were supported on a frictionless maglev, could the maglev power be varied at specific locations to offset any off-axis forces (including those caused by moving people, equipment, vehicles, fluids) in real time?
« Last Edit: 03/09/2015 08:47 PM by punder »

#### scienceguy

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##### Re: Artificial gravity habitat on a planetary body
« Reply #3 on: 03/09/2015 08:59 PM »
that's a good idea
e^(pi*i) = -1

#### DanielW

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##### Re: Artificial gravity habitat on a planetary body
« Reply #4 on: 03/09/2015 09:51 PM »
All the ladies worth having will immediately inquire if that is 1000 lbf or 1000 lbm and quietly think to themselves "imperial units? this far from England?"

#### Paul451

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##### Re: Artificial gravity habitat on a planetary body
« Reply #5 on: 03/10/2015 01:58 AM »
or else have to deal with continual twisting forces in the bearings.

Meh, the twisting-acceleration on the axis due to precession is 0.05% of 1g at Ceres' equator¹. (Can't work out the actual torque-force without knowing the angular momentum of the structure. But in linear terms, it's around a pound of force per ton-equivalent of mass. Your shaft will need to handle more than that for normal vibration and mass changes from people walking around.)

[At the moon's equator¹, it's just 4 billionths of 1g. Rounding off error's rounding off error.]

However, IMO, you wouldn't hang your structure from a central boom-arm, you'd support it on a ring-track. That opens up your maximum radius beyond the structural strength of the support arm. As a side effect, that track could deal with a huge amount of torque force, because it's then compressive on the rim, not twisting on the axis.

¹ Equator is the worst case. Force drops as you increase latitude.
« Last Edit: 03/10/2015 02:02 AM by Paul451 »

#### HMXHMX

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##### Re: Artificial gravity habitat on a planetary body
« Reply #6 on: 03/10/2015 03:03 AM »
MORE artificial gravity!  WooHoo!!      Apologies if this has been done before on this site or elsewhere.  Please send me the link if so.

I did some math to come up with the "floor shape" for a spinning AG habitat sitting on a planetary surface.  By floor shape, I mean the curve that enables someone to stand orthogonal to the surface and not feel any (significant) tugs to the side.

(It's not a hard problem--would make a good homework question for a calculus-based freshman physics class--but it was difficult for me, anyway.  It was also a lot of fun exploding the heads of myself and a few friends as we whiteboarded it out.)

The curve is a parabola, y = (w2/2Ag)r2 where:

w = angular velocity of the habitat
Ag = the planet's gravitational acceleration
r = distance from the hab's center of rotation.

You might ask, why the heck would you even do that?  Well, it might turn out that lunar gravity, for instance, isn't enough to support normal bone and muscle health.  Or, adults might be fine, but children--either gestating or growing up--might need more gee to develop correctly.  So there might be a reason to do it, or there might not.  Either way, the math and physics are interesting.

I’m doing a single-variable integration, setting w and Ag as constants.  Someone more edumacated might come up with a more general solution in which these are variables.

Here goes:

At any point on the floor, there are two vectors: Ag and centripetal acceleration Ac.
Ac = rw2.
The normal (to the floor) acceleration An is the sum of Ac and Ag.  This is the “target” acceleration, most likely 9.81m/s2.  Its slope is Ag/Ac.
The slope of the “floor curve” is perpendicular to this vector, so it is Ac/Ag = rw2/Ag.
Since you now have a slope, you can integrate to get the curve.  The integral of y’ = rw2/Ag is y = (w2/2Ag)r2.

That's it!  Now you can write an equation for any combination of angular velocity and local gravity to design “floors” for rotating habitats on the Moon, Mars, or any other planetary body.

For example, say you’d like a 1rpm habitat on the Moon, with a max “gravity” equal to Earth’s, 9.81m/s2.

Ac = sqt(9.81m/s2 - 1.63m/s2) = 9.67m/s2
r = Ac/w2 = 9.67m/s2/(2pi/60s)2 = 882m
y = (w2/2Ag)r2 = ((pi/30)2/1.63)r2
y = 0.003365r2
Subbing in, you get y = 2618m.

Now say your rotating pressure vessel is a torus, and the cross sectional diameter is such that the “bottom” point on the floor (min r) is 20m below the “top” (max r).

rmin = sqt(2598/.003365) = 879m
Acmin = (879m)((pi/30)2) = 9.64m/s2
Anmin = sqt((9.64m/s2)2 + (1.63m/s2)2) = 9.78m/s2

Again: if you see a horrible math or physics mistake (or a sup/sub problem or typo amounting to the same thing), please let me know.  Don’t be shy.      A diagram is attached to help visualize it.

First time this was proposed, to my knowledge, was by Peter Diamandis in:  http://ssi.org/ssi-conference-abstracts/space-manufacturing-6/ in 1988.

#### punder

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##### Re: Artificial gravity habitat on a planetary body
« Reply #7 on: 03/10/2015 02:30 PM »
First time this was proposed, to my knowledge, was by Peter Diamandis in:  http://ssi.org/ssi-conference-abstracts/space-manufacturing-6/ in 1988.

Knew that was inevitable.  Will check the link--hopefully the math is included so I can confirm my stuff is correct.

Well heck, I'm in good company!

If anyone here would be kind enough to check my math independently, I'd be much obliged.

Edit, is this it?  Any way I can get a copy?

Reconsidering Artificial Gravity for Twenty-first Century Space Habitats/Peter H. Diamandis, M.I.T., Harvard Medical School, pg. 55

Abstract: The paper examines the medical rationale for artificial gravity and reviews past investigations into the optimal rotation velocity and habitat radius. The paper also proposes new directions for research in the field of Artificial Gravity.
« Last Edit: 03/10/2015 02:55 PM by punder »

#### punder

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##### Re: Artificial gravity habitat on a planetary body
« Reply #8 on: 03/10/2015 02:49 PM »
or else have to deal with continual twisting forces in the bearings.

Meh, the twisting-acceleration on the axis due to precession is 0.05% of 1g at Ceres' equator¹. (Can't work out the actual torque-force without knowing the angular momentum of the structure. But in linear terms, it's around a pound of force per ton-equivalent of mass. Your shaft will need to handle more than that for normal vibration and mass changes from people walking around.)

[At the moon's equator¹, it's just 4 billionths of 1g. Rounding off error's rounding off error.]

However, IMO, you wouldn't hang your structure from a central boom-arm, you'd support it on a ring-track. That opens up your maximum radius beyond the structural strength of the support arm. As a side effect, that track could deal with a huge amount of torque force, because it's then compressive on the rim, not twisting on the axis.

¹ Equator is the worst case. Force drops as you increase latitude.

Maglev ring-track seems like a really good idea, because it might also counteract other forces as I noted above.

Without a hub, the method of entry/exit gets interesting.  For a lunar 1rpm/1g hab, the tangential velocity is 92 m/s, or 207 mph.  Construct a roadway (or maglev line) that starts off tangent to the rim, then curves into a precisely banked track around the hab.  Your mooncar (or maglev pod) accelerates up the track until it matches speed with a grapple mechanism extended from the rim, and the grapple lifts you into a garage which is either an airlock itself, or has an airlock entrance or docking tube for personnel.

To get out of the hab, the maglev pod seems like a better idea--or else your mooncar wheels better be spinning pretty fast before rubber meets the road!

#### AlanSE

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##### Re: Artificial gravity habitat on a planetary body
« Reply #9 on: 03/10/2015 04:40 PM »
I think that the requirements (and thus, operation) for the moon become pretty self-obvious.

Instead of mucking with a tremendously complex onload and offload system, you would just run it for 1-2 hours and then stop it to change passengers... like an amusement park ride. It's the same thing after all. Children will be required to have more sessions on it than adults, but it wouldn't consume their entire day, every day.

We spend plenty of time lying down as things are right now. I've never heard of a good cardiovascular argument of the importance of the gravity gradient going front-to-back, only head-to-toe. NASA has studied people by paying them to lie down for a straight month. They get negative health effects, just like our astronauts.

Also, tensile members have a lot of advantages over compressive members. For all its time operating, you would want it held in place by tethers attached at a central hub on a lattice tower. However, to keep it suspended after it stops with this system would entail adding compressional members. For that reason, you would want it to rest against the ground after it stops. So I would envision a hybrid system that has

1. A tower/tether that holds it in place for the spin and
2. A track that it rests on to stop and only imparts force normally

The most foolproof way to keep the track's force always normal would be to have it act perpendicular to the tether. Like this:

This isn't glamorous, but I think it pretty clearly satisfies the requirements. I have a hard time developing a good argument to change any part of it.

EDIT: Upon reflection, those proportions would be horribly inaccurate for the moon, maybe for Mars that image would be vaguely accurate. The angle for the moon would be very low. You would probably use a conic mound instead of a skeleton tower.

Design crossover point

Consider a station on the near-earth asteroid of Amun. Its gravity isn't a few percent of 1g, it's much lower than that. For this case, it would seem wasteful to use bearings to move the station's axis of rotation along with the rotation of the asteroid. Nonetheless, you still might want to use it for shielding and have something which slightly resembles a "surface" station for that reason.

I would picture something more like a Bernal Sphere, and its is located inside a spherical cavity in the rock. The axis of rotation precesses as the asteroid rotates, similar to http://www.crystalinks.com/precession.gif. Granted, you would probably avoid this by placing it at the pole or equator.

As we look at larger masses, at some point this becomes unreasonable. 1 rotation per minute means that the body's gravity would flip relative to your ground that often. 1% change in gravity, applied over a full vertical circle, is enough to knock some things off your table. So if the body's gravity was large enough, you would make it like the moon example. If it was small enough, you would make it more like a stand-alone O'Neil colony. It's interesting to consider that the best option would probably depend on the size of the body you build it on, and also the size of the wheel itself.
« Last Edit: 03/10/2015 04:51 PM by AlanSE »

#### JasonAW3

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##### Re: Artificial gravity habitat on a planetary body
« Reply #10 on: 03/10/2015 05:13 PM »
or else have to deal with continual twisting forces in the bearings.

Meh, the twisting-acceleration on the axis due to precession is 0.05% of 1g at Ceres' equator¹. (Can't work out the actual torque-force without knowing the angular momentum of the structure. But in linear terms, it's around a pound of force per ton-equivalent of mass. Your shaft will need to handle more than that for normal vibration and mass changes from people walking around.)

[At the moon's equator¹, it's just 4 billionths of 1g. Rounding off error's rounding off error.]

However, IMO, you wouldn't hang your structure from a central boom-arm, you'd support it on a ring-track. That opens up your maximum radius beyond the structural strength of the support arm. As a side effect, that track could deal with a huge amount of torque force, because it's then compressive on the rim, not twisting on the axis.

¹ Equator is the worst case. Force drops as you increase latitude.

Dang!  I was thinking about an Equitorial Mag Lev train. (ring, whatever)

Come to think of it, one could build a ring around Ceres, attach it just enough to maintain position relative to the asteroid.  use the asteroid to land or moor craft to and use a surface, maglev train to spiral up to the ring, flipping over bofore contact with the ring. Once on the ring, the "Train" would shunt to oneside to avoid impacting the spiral ramp.

The habitat ring itself could grow to gigantic proportions.  Ceres being about 960 Kilometers in diameter would allow a ring of, say, 10 miles width by 10 miles from inner ring to outer surface, with 5 miles of levels from the outer surface to the inner "Atrium".  The inner "Atrium" would have an inner surface area at the 5 mile point of over 7,500,000 Kilometers.  Tens of millions of people could live in a structure like that, and farm more than sufficent crops as well.

The slope of the ring would be such that at worst, it would seem like living in a gently rolling valley.

Unless built with automated systems and modular, like current ship building yards do, it would take generations to build such a structure, as it is, it would still take a few decades. But it could be well worth it as the "Gateway to the Outer Solar System".  It could become a major trading hub.
My God!  It's full of universes!

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##### Re: Artificial gravity habitat on a planetary body
« Reply #11 on: 03/10/2015 05:16 PM »
That would massively reduce the mass and complexity that has to be transported/built. If a short while a day in 1 (or more) g is enough, and astronauts are trained to withstand a few rpm, the structure becomes reasonably small. And safer than a similar design in microgravity.

#### JasonAW3

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##### Re: Artificial gravity habitat on a planetary body
« Reply #12 on: 03/10/2015 07:27 PM »
Here's another thought;

Build a ring habitat as a subway around the circumfrence of Ceres.  Putting it subsurface provides radiation protection as well as meteor and micrometeor protection.

Construction would be somewhat easier and the materials,mostly water, could be used for a colony.
My God!  It's full of universes!

#### Nilof

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##### Re: Artificial gravity habitat on a planetary body
« Reply #13 on: 03/10/2015 07:42 PM »
The fact that the "floor shape" is a parabola is incidentally something that has lead to some lunar space telescope proposals. I recall hearing about a lunar telescope proposal that would be essentially a rotating platter of mercury.
For a variable Isp spacecraft running at constant power and constant acceleration, the mass ratio is linear in delta-v.   Δv = ve0(MR-1). Or equivalently: Δv = vef PMF. Also, this is energy-optimal for a fixed delta-v and mass ratio.

#### punder

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##### Re: Artificial gravity habitat on a planetary body
« Reply #14 on: 03/10/2015 08:15 PM »
The fact that the "floor shape" is a parabola is incidentally something that has lead to some lunar space telescope proposals. I recall hearing about a lunar telescope proposal that would be essentially a rotating platter of mercury.

Some have been built on Earth!  http://en.wikipedia.org/wiki/Liquid_mirror_telescope

#### Paul451

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##### Re: Artificial gravity habitat on a planetary body
« Reply #15 on: 03/10/2015 11:24 PM »
On the surface:
[...]
1. A tower/tether that holds it in place for the spin and
2. A track that it rests on to stop and only imparts force normally
The most foolproof way to keep the track's force always normal would be to have it act perpendicular to the tether. Like this:

Don't see the point of the tower. Tensile cables running across the centre would achieve the same result. There's nothing that required the cables to be parallel to the plane of artificial gravity.

However, more importantly, I can't see why you would bank the modules/track outwards like that. Instead, place your tensile structure in a ring around the outside of the modules. Angle the modules inwards at the appropriate floor-angle. Have the overall structure supported by tracks on a compressive foundation under the ring. Done.

[top diagram in my vague effort at the bottom of this comment]

Inside an asteroid:
For this case, it would seem wasteful to use bearings to move the station's axis of rotation along with the rotation of the asteroid. Nonetheless, you still might want to use it for shielding and have something which slightly resembles a "surface" station for that reason.

Again, use a simple torus tensile structure. The force of precession is so low that a guide-track would be a fairly light structure. Gravity is so low that essentially it's a free-flying toroidal space-station, with an extra guide-track to keep it centred in its ring-tunnel.

Anything else is overkill.

[bottom diagram in my vague effort at the bottom of this comment]

As we look at larger masses, at some point this becomes unreasonable. 1 rotation per minute means that the body's gravity would flip relative to your ground that often. 1% change in gravity, applied over a full vertical circle, is enough to knock some things off your table.

You simply align the spin axis of the toroid towards the centre of mass, problem solved. As I said, precession forces are trivial.

you would just run it for 1-2 hours and then stop it to change passengers...

The point of a spin-structure isn't just health effects, it's also simplifying the engineering of every other system on the station. Having a hab that is required to work under spin, and during spin-up/down, and under micro-g, would make the engineering more complex.

Whenever you read astronaut logs from ISS, you are struck by how ridiculously inconvenient micro-g is for doing anything complex. Not being able to go into a cargo capsule safely because little shavings of metal may be floating around (ready to go in your eye or lungs) that were completely harmless and unnoticed when the capsule was on the ground. Having the ECLSS filters collect of every single thing that gets airborne, because in micro-g vents become the "lowest" point. Having nothing quite work like it was designed because tiny effects (like surface tension) become dominant in micro-g.

Once you scale up a habitat beyond a certain point, you won't want to deal with all that crap.

Aside:
Quote

You can use {url=http://www.crystalinks.com/precession.gif}{img}http://www.crystalinks.com/precession.gif{/img}{/url} to create a visible embedded image with a clickable link. (Subbing square brackets back for the curlies, obviously.)

Such as...

--

My own aforementioned vague effort should appear below:

Greenish-rings are the tensile structure. Black in the asteroid represents the cavity inside the asteroid (and an access tunnel to the surface). Black on the lunar-surface is obviously the track; it doesn't need to be banked inwards because there's no net outwards force, the tensile structure absorbs that. (Indeed, the wheels wouldn't be angled either, but that looked weird.)
« Last Edit: 03/10/2015 11:29 PM by Paul451 »

#### Paul451

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##### Re: Artificial gravity habitat on a planetary body
« Reply #16 on: 03/10/2015 11:58 PM »
Come to think of it, one could build a ring around Ceres, attach it just enough to maintain position relative to the asteroid.  use the asteroid to land or moor craft to and use a surface, maglev train to spiral up to the ring, flipping over bofore contact with the ring. Once on the ring, the "Train" would shunt to oneside to avoid impacting the spiral ramp.

Not sure what you're describing here. Do you mean a ring around the equator that is spun for AG? (Rotating at 2km/s.) So "down" is outwards from Ceres, "up" is inwards? Seems deeply (deeply) overkill, even if there were enough people to justify it.

Re: Giant spin cities in general.
IMO, these are a bit like an arcology on Earth. It might be technologically possible, but it's impossible for such a singular structure to develop in practice. Instead, just as our cities are made up of many smaller buildings sharing a common infrastructure, I would expect many smaller stand-alone spin-habitats sharing a common infrastructure on Ceres. (Probably underground for shielding (or hopefully, under-ice.)) This one is the equivalent of an apartment block, this one is an office, this one one is a factory... Each set to their own preferred gravity levels (or lack thereof).

#### TrueBlueWitt

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##### Re: Artificial gravity habitat on a planetary body
« Reply #17 on: 03/11/2015 12:14 AM »
In the already low atmospheric pressure of Mars or lack there of on Ceres..
Couldn't you use pretty thin walls(Inflatable?) for a Variable G Hyperloop Ring?

You think Elon came up with that for Earth use first?
« Last Edit: 03/11/2015 12:19 AM by TrueBlueWitt »

#### AlanSE

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##### Re: Artificial gravity habitat on a planetary body
« Reply #18 on: 03/11/2015 01:28 AM »
For this case, it would seem wasteful to use bearings to move the station's axis of rotation along with the rotation of the asteroid. Nonetheless, you still might want to use it for shielding and have something which slightly resembles a "surface" station for that reason.

Again, use a simple torus tensile structure. The force of precession is so low that a guide-track would be a fairly light structure. Gravity is so low that essentially it's a free-flying toroidal space-station, with an extra guide-track to keep it centred in its ring-tunnel.

Anything else is overkill.

Let me bring up one specific topic here: At what set of parameters does the force of precession (I'll say "acceleration") become significant?

Is it insignificant in all possible scenarios? I don't believe that at all. Also, is the acceleration due to precession felt as a constant to the passengers? Or does it vary with angular position?

I'm starting to come to the thinking that the acceleration due to precession would vary back and fourth sinusoidally - left and right to the people there. That's a problem, because even small accelerations can wreck havoc on people's sense of motion. I think being jostled 0.1 g left and right might be utterly intolerable.

So when does this happen? I'm leaning toward the following order-of-magnitude estimation: whenever the period of rotation of the artificial gravity is similar to the period of rotation of the asteroid or moon. So let's say the day-length is 6 hours. If the rotation of the habitat is 1/10th of that, then the precession acceleration is probably too high. That would be a rotation once every 36 minutes, which would imply an enormous artificial gravity radius. However, if the criteria was 1/100th, this starts to look more like actual designs.

While this doesn't appear to be restrictive for most scenarios, we still need to get a good grasp on why or why not.

#### Alf Fass

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##### Re: Artificial gravity habitat on a planetary body
« Reply #19 on: 03/11/2015 02:24 AM »
I mentioned the idea here, but my math isn't good enough to cover it in the the detail you have.
http://forum.nasaspaceflight.com/index.php?topic=35225.msg1230796#msg1230796
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