# Spin gravity and atmospheric retention

#### bilby

##### Fair dinkum thinkum
So, I am reading my usual shit sci-fi, and one of the protagonists is on a rotating space station. You know, the 2001 type that generates gravity in a wheel-shaped facility by spinning.

Terrorists are threatening the facility. They plan to destroy the integrity of the hub, and kill everyone - because the air will all rapidly be drawn into the vacuum of space.

They're not able to get to the rim; so any damage they do is limited to the hub - but they have managed to disable any airtight seals and doors in the spokes.

Our hero speculates that the spin gravity might prevent the loss of all of the atmosphere, if the terrorists do explode their bomb, and depressurise the hub.

So my questions are:

1) Just how big (or how rapidly rotating) would such a rotating space station need to be, to retain sufficient atmosphere to keep its inhabitants alive, despite the station being opened to space at the hub? (And is this possible for a small [say, less than 10km diameter wheel] with reasonable maximum gravity at the rim - say 0.8 to 1.2g).

2) If the station were big (or fast) enough, wouldn't that imply that workers at the hub of such a station would be working in a vacuum, or at least at dangerously low pressures requiring protective suits, in normal circumstances (eg not a terrorist attack), or the inhabitants of the rim would be living at very high pressures, requiring low nitrogen 'air', like that used by divers, unless the spokes had airlocks or other seals to prevent the hub atmosphere from draining into tne rim?

#### Jokodo

##### Veteran Member
So, I am reading my usual shit sci-fi, and one of the protagonists is on a rotating space station. You know, the 2001 type that generates gravity in a wheel-shaped facility by spinning.

Terrorists are threatening the facility. They plan to destroy the integrity of the hub, and kill everyone - because the air will all rapidly be drawn into the vacuum of space.

They're not able to get to the rim; so any damage they do is limited to the hub - but they have managed to disable any airtight seals and doors in the spokes.

Our hero speculates that the spin gravity might prevent the loss of all of the atmosphere, if the terrorists do explode their bomb, and depressurise the hub.

So my questions are:

1) Just how big (or how rapidly rotating) would such a rotating space station need to be, to retain sufficient atmosphere to keep its inhabitants alive, despite the station being opened to space at the hub? (And is this possible for a small [say, less than 10km diameter wheel] with reasonable maximum gravity at the rim - say 0.8 to 1.2g).

2) If the station were big (or fast) enough, wouldn't that imply that workers at the hub of such a station would be working in a vacuum, or at least at dangerously low pressures requiring protective suits, in normal circumstances (eg not a terrorist attack), or the inhabitants of the rim would be living at very high pressures, requiring low nitrogen 'air', like that used by divers, unless the spokes had airlocks or other seals to prevent the hub atmosphere from draining into tne rim?

Earth's atmosphere reaches one half of sea level pressure at an altitude of about 5500 meters, and a quarter at 11000 meters. However compared to earth's gravity, acceleration diminishes much more rapidly in a rotating wheel, so the atmosphere at the hub of a11km diameter wheel will be denser than at 5500 on earth. So the air loss would definitely be significant, though slower than with a hole in the rim.

#### Loren Pechtel

##### Super Moderator
Staff member
So, I am reading my usual shit sci-fi, and one of the protagonists is on a rotating space station. You know, the 2001 type that generates gravity in a wheel-shaped facility by spinning.

Terrorists are threatening the facility. They plan to destroy the integrity of the hub, and kill everyone - because the air will all rapidly be drawn into the vacuum of space.

They're not able to get to the rim; so any damage they do is limited to the hub - but they have managed to disable any airtight seals and doors in the spokes.

Our hero speculates that the spin gravity might prevent the loss of all of the atmosphere, if the terrorists do explode their bomb, and depressurise the hub.

So my questions are:

1) Just how big (or how rapidly rotating) would such a rotating space station need to be, to retain sufficient atmosphere to keep its inhabitants alive, despite the station being opened to space at the hub? (And is this possible for a small [say, less than 10km diameter wheel] with reasonable maximum gravity at the rim - say 0.8 to 1.2g).

2) If the station were big (or fast) enough, wouldn't that imply that workers at the hub of such a station would be working in a vacuum, or at least at dangerously low pressures requiring protective suits, in normal circumstances (eg not a terrorist attack), or the inhabitants of the rim would be living at very high pressures, requiring low nitrogen 'air', like that used by divers, unless the spokes had airlocks or other seals to prevent the hub atmosphere from draining into tne rim?

Hero needs to quit speculating. It's not going to happen. The atmosphere in the station is going to drop off somewhat slower than it does on Earth as the "gravity" drops off more slowly.

That being said, if the terrorists can kill everyone by venting the hub the engineers should be executed for incompetence. Even the ISS has airtight doors!

#### bilby

##### Fair dinkum thinkum
So, I am reading my usual shit sci-fi, and one of the protagonists is on a rotating space station. You know, the 2001 type that generates gravity in a wheel-shaped facility by spinning.

Terrorists are threatening the facility. They plan to destroy the integrity of the hub, and kill everyone - because the air will all rapidly be drawn into the vacuum of space.

They're not able to get to the rim; so any damage they do is limited to the hub - but they have managed to disable any airtight seals and doors in the spokes.

Our hero speculates that the spin gravity might prevent the loss of all of the atmosphere, if the terrorists do explode their bomb, and depressurise the hub.

So my questions are:

1) Just how big (or how rapidly rotating) would such a rotating space station need to be, to retain sufficient atmosphere to keep its inhabitants alive, despite the station being opened to space at the hub? (And is this possible for a small [say, less than 10km diameter wheel] with reasonable maximum gravity at the rim - say 0.8 to 1.2g).

2) If the station were big (or fast) enough, wouldn't that imply that workers at the hub of such a station would be working in a vacuum, or at least at dangerously low pressures requiring protective suits, in normal circumstances (eg not a terrorist attack), or the inhabitants of the rim would be living at very high pressures, requiring low nitrogen 'air', like that used by divers, unless the spokes had airlocks or other seals to prevent the hub atmosphere from draining into tne rim?

Hero needs to quit speculating. It's not going to happen. The atmosphere in the station is going to drop off somewhat slower than it does on Earth as the "gravity" drops off more slowly.

That being said, if the terrorists can kill everyone by venting the hub the engineers should be executed for incompetence. Even the ISS has airtight doors!

For sure. It's shit sci-fi; The stuff the author doesn't want to be vulnerable to terrorists is made from unobtainium, which couldn't be used in places that the plot requires to be vulnerable, because technobabble.

#### Jimmy Higgins

##### Contributor
The movie Elysian had an exposed atmosphere in a rotating ring, permanently. Same premise, as the air would hold within the ring due to the rotation. I do wonder if the speed for 1g gravity would be good enough for holding in the atmosphere.

#### Worldtraveller

##### Veteran Member
The movie Elysian had an exposed atmosphere in a rotating ring, permanently. Same premise, as the air would hold within the ring due to the rotation. I do wonder if the speed for 1g gravity would be good enough for holding in the atmosphere.
Sort of. The gravity differential is gets worse, the smaller the ring is. Air would vent out almost as quickly by an opening in the hub as it would in a stationary room at 1g. Pretty quickly, but not quite as fast as it would in zero g. The pressure differential between the presumably normal atmosphere and a vacuum is much higher than anything that would be overcome by the artificial gravity.

Even on a place like Mars, with ~1/3G, if we started with an atmosphere at the same pressure as what's on Earth, it would 'only' last about 30k years (IIRC) without some means to constantly renew it. The moon would last about 10k years (again, going from memory).

#### bilby

##### Fair dinkum thinkum
The movie Elysian had an exposed atmosphere in a rotating ring, permanently. Same premise, as the air would hold within the ring due to the rotation. I do wonder if the speed for 1g gravity would be good enough for holding in the atmosphere.
Sort of. The gravity differential is gets worse, the smaller the ring is. Air would vent out almost as quickly by an opening in the hub as it would in a stationary room at 1g. Pretty quickly, but not quite as fast as it would in zero g. The pressure differential between the presumably normal atmosphere and a vacuum is much higher than anything that would be overcome by the artificial gravity.

Even on a place like Mars, with ~1/3G, if we started with an atmosphere at the same pressure as what's on Earth, it would 'only' last about 30k years (IIRC) without some means to constantly renew it. The moon would last about 10k years (again, going from memory).

Thanks.

Presumably, to hold an atmosphere at survivable pressures for the really long term would need an enormous open ring - something on the scale of Niven's Ringworld

#### Treedbear

##### Veteran Member
The movie Elysian had an exposed atmosphere in a rotating ring, permanently. Same premise, as the air would hold within the ring due to the rotation. I do wonder if the speed for 1g gravity would be good enough for holding in the atmosphere.

It's not only a matter of holding the air inside the ring. It needs to approximate the air pressure on Earth. Googling the weight of air I found it's .0807 pounds/cu ft. So 15 pounds/sq in. divided by .0807 pounds/(12 x 12 x 12) cubic inches and then dividing by 12 to get feet and then by 5280 to get miles I get a ring width of 5 miles. Elysium seemed to be too shallow. At least in the movie, according this:
From the production crew, we know that the space station Elysium is 60 kilometers, or about 37.3 miles, in diameter and about 2 kilometers (1.2 miles) thick.

#### Loren Pechtel

##### Super Moderator
Staff member
The movie Elysian had an exposed atmosphere in a rotating ring, permanently. Same premise, as the air would hold within the ring due to the rotation. I do wonder if the speed for 1g gravity would be good enough for holding in the atmosphere.
Sort of. The gravity differential is gets worse, the smaller the ring is. Air would vent out almost as quickly by an opening in the hub as it would in a stationary room at 1g. Pretty quickly, but not quite as fast as it would in zero g. The pressure differential between the presumably normal atmosphere and a vacuum is much higher than anything that would be overcome by the artificial gravity.

Even on a place like Mars, with ~1/3G, if we started with an atmosphere at the same pressure as what's on Earth, it would 'only' last about 30k years (IIRC) without some means to constantly renew it. The moon would last about 10k years (again, going from memory).

Thanks.

Presumably, to hold an atmosphere at survivable pressures for the really long term would need an enormous open ring - something on the scale of Niven's Ringworld

Exactly. Niven handwaved when he needed to but tried to keep the numbers right if he could. 1000 mile walls are what it takes if you want the top open.

The key to understanding atmosphere is scale height. It's the distance where the pressure drops to 1/e of what it was. For Earth it's about 8.5km at surface temperature and gravity. Thus 100km up the atmosphere has dropped to about 1/(e^12) of what it is at the surface. That's 6 parts in a million of sea level--about half a pascal. Over time that would spill a lot of air. You need to go a lot higher to hold an atmosphere for a long time.

#### Jokodo

##### Veteran Member
Thanks.

Presumably, to hold an atmosphere at survivable pressures for the really long term would need an enormous open ring - something on the scale of Niven's Ringworld

Exactly. Niven handwaved when he needed to but tried to keep the numbers right if he could. 1000 mile walls are what it takes if you want the top open.

The key to understanding atmosphere is scale height. It's the distance where the pressure drops to 1/e of what it was. For Earth it's about 8.5km at surface temperature and gravity. Thus 100km up the atmosphere has dropped to about 1/(e^12) of what it is at the surface. That's 6 parts in a million of sea level--about half a pascal. Over time that would spill a lot of air. You need to go a lot higher to hold an atmosphere for a long time.

I'm pretty sure it's 5.5, not 8.5 km where three pressure is half its sea levelvalue. Eg here http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/prs/hght.rxml 8.5 sounds like the kind of figure you get by converting 5.5 miles to km, only that it's already in km, and can be avoided bu using metric units exclusively and consistently.

That's on earth. In a spinning wheel, the drop in "gravity" (acceleration) would be more significant (I.e. half at half the radius) and the drop of in pressure this slower.

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#### Loren Pechtel

##### Super Moderator
Staff member
Thanks.

Presumably, to hold an atmosphere at survivable pressures for the really long term would need an enormous open ring - something on the scale of Niven's Ringworld

Exactly. Niven handwaved when he needed to but tried to keep the numbers right if he could. 1000 mile walls are what it takes if you want the top open.

The key to understanding atmosphere is scale height. It's the distance where the pressure drops to 1/e of what it was. For Earth it's about 8.5km at surface temperature and gravity. Thus 100km up the atmosphere has dropped to about 1/(e^12) of what it is at the surface. That's 6 parts in a million of sea level--about half a pascal. Over time that would spill a lot of air. You need to go a lot higher to hold an atmosphere for a long time.

I'm pretty sure it's 5.5, not 8.5 km where three pressure is half its sea levelvalue. Eg here http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/prs/hght.rxml 8.5 sounds like the kind of figure you get by converting 5.5 miles to km, only that it's already in km, and can be avoided bu using metric units exclusively and consistently.

That's on earth. In a spinning wheel, the drop in "gravity" (acceleration) would be more significant (I.e. half at half the radius) and the drop of in pressure this slower.

Note that I am talking about 1/e, not 1/2. The dropoff of "gravity" on a Ringworld is much slower than on Earth. I do agree on a spinning wheel the "gravity" drops off much faster, but building a wheel big enough to have vacuum at the hub would be hard!

#### lpetrich

##### Contributor
To settle this question, let us do some hydrodynamics. I will do static and quasi-static hydrodynamics only, to make it easier.

First, the pressure-equilibrium equation.
$${\vec \nabla} P = {\vec f} = \rho {\vec a} = - \rho {\vec \nabla} V$$
where P is the pressure, f is the force density, (rho) is the density, a is the acceleration, and V is the potential.

For small distances h and acceleration of gravity g,
$$V = g h$$

For centrifugal acceleration at distance r from a central axis with angular velocity w,
$$V = - \frac12 \omega^2 r^2$$

Density is related to pressure, temperature T, and molecular weight m by the ideal gas law:
$$P = \frac{\rho k T}{m}$$
Non-ideality is usually small enough to ignore without loss of much accuracy.

The pressure-equilibrium equation and the ideal gas law are two equations for three quantities, (rho), P, and T, so we need a third equation.

#### lpetrich

##### Contributor
I will consider two types, a power-law adiabatic,
$$P \sim \rho^\gamma$$
and isothermal, T = constant.

Adiabatic first, since that is essentially convective equilibrium, equilibrium over moving up and down. It is easiest to find the temperature, and from the power law, we have
$$\rho \sim T^{1/(\gamma-1)} ,\ P \sim T^{\gamma/(\gamma-1)}$$

Plugging it into the hydrostatic-equilibrium equation gives us
$$\frac{\gamma}{\gamma-1} \frac{k}{m} {\vec \nabla} T = - {\vec \nabla} V$$
giving us
$$T = T_0 - \frac{\gamma-1}{\gamma} \frac{m}{k} (V - V_0)$$

For the gravity case, we get
$$T = T_0 - \Gamma (h - h_0) ,\ \Gamma = \frac{\gamma-1}{\gamma} \frac{mg}{k}$$
(Gamma) is called the "adiabatic lapse rate".

The Earth's dry adiabatic lapse rate is 9.8 C / km.

One can do similar calculations for the centrifugal case, and the center-to-rim temperature difference is half that for constant acceleration = rim acceleration. Thus, if one has 1 g at the rim, then temperature difference between the center and the rim is 4.9 C / km * (radius).

So there won't be very much difference unless the habitat is very large (radius ~ several kilometers).

#### lpetrich

##### Contributor
Now the isothermal case. It has
$$\frac{kT}{m} {\vec \nabla} \rho = - \rho {\nabla V}$$
giving us
$$\rho = \rho_0 \exp \left( - \frac{mV}{kT} \right)$$
and
$$P = P_0 \exp \left( - \frac{mV}{kT} \right)$$

Here also, detailed calculation does not show much change in air density unless the habitat is very large.

#### lpetrich

##### Contributor
Returning to the OP's conundrum, if the hub gets an air leak, the entire station's air will leak out unless the station is divided into sealable compartments. For a large station, I think that such compartments will be a necessity.

#### bilby

##### Fair dinkum thinkum
Returning to the OP's conundrum, if the hub gets an air leak, the entire station's air will leak out unless the station is divided into sealable compartments. For a large station, I think that such compartments will be a necessity.

I suspected that might be the case. A small (ie less than tens of km diameter) space station is essentially at similar risk of decompression regardless of spin gravity - the rotation helps people at the rim to live in earthlike gravity, but does little to protect them from any kind of serious atmosphere leak, even at the hub.

Which is probably a good thing in the absence of a leak, as it allows workers at the hub to breathe without spacesuits. If an uncontained hull breach at the hub was a minor problem for those at the rim, then working unprotected at the hub would be impossible.