Air pressure calculation at an arbitrary point within a sphere

[writtenword]

Well my first post went into moderation, I suspect because I included the links for the locations of my source information and I’m sure they want to make sure I am not spamming or advertising or something nefarious. Perhaps I will take a different tack this time. No linking of reference material, so I guess you will just have to trust me.

I am trying to calculate the air pressure of an arbitrary point within a sphere filled with normal air. I had not imagined that straightforward calculations of air pressure would be so hard to find but it has been a beast, a well hidden beast.

I have the density of the atmosphere in the sphere and we can call the temperature 22 C or 295.15 K.

I found the below formula at a very secret and entirely undisclosed location -

D=P/R*T
D = density, kg/m3
P = pressure, Pascals ( multiply mb by 100 to get Pascals)
R = specific gas constant , J/(kg*degK) = 287.05 for dry air
T = temperature, deg K = deg C + 273.15
(101325 Pascals at sea level for earth)

And while it is a very pretty equation I am not trying to solve for density, I know the density via volume and mass. I want the pressure of a given point.
So I need to juggle the equation a little right? I need to make it

P=D*R*T

Is that right? I have no math brain for inverse properties.

Also

I should have mentioned earlier that this sphere filled with air has mass and thus gravity, so the air is not constant through-out but acted upon by gravity, denser toward the center and thinner toward the shell. But we can solve this by subtracting volume below us (relative to the center) from the total volume and recalculating density at our arbitrary point.

So my question is:

In order to make D=P/R*T solve for P is the new formula P=D*R*T?

If there is a better way to calculate air pressure at an arbitrary point as described above, I would love to know it

 

[Simon Bridge]

You want the pressure profile of a self-gravitating gas sphere in hydrostatic equilibrium? Or do you mean there to be some positive pressure for the shell to hold in?

This will have to be a big volume of gas and it's not a simple calculation.
http://www.ita.uni-heidelberg.de/~dullemond/lectures/starplanet_2005/chap_05_cloudcores.pdf
... third slide gives you the equations. You can modify the equilibrium condition to account for a pressure container if you like.

 

[writtenword]

Huh, that is a good darn find I must say. While my model is not a self gravitating gas sphere per se, at 1 AU in diameter it could be I suppose after a fashion.

I had envisioned it as a volume of gas contained in a sphere such that it’s mass and volume does not have to correlate to its gravity for cohesion as a sphere. So I can increase or decrease the density of the air inside the sphere affecting its mass without concern for losing its gravity based existence. There is no need for equilibrium between gravity and density.

It looks like the formulae referenced here are indeed complicated by the need to describe the shifting relationship between gravity and pressure in relation to its volume.

I would like to consider the jury still out on this and I think that lack of link between gravity and density might simplify the equation, its core computation still evades me. I think if I can just get a decent descriptive equation of air pressure at a given depth in the sphere, I can work out all the other calculations to arrive at my final goal of a macro-construction supporting a habitable zone, a central point of gravity and the numbers to chart its existence.

Speculatively speaking, it’s kind of a cool thought.

Thanks for the link Simon, I will definitely spend some time in those slides, the info is interesting on its own.

 

[Simon Bridge]

You cannot avoid the link between gravity and density.

All masses are self gravitating - with nothing to balance this, even a 1m sphere of gas will contract to a solid. Your gas ball has a temperature so the KE of the molecules is providing the push outwards. If you propose a net outwards pressure so you need a container for it, then the temperature has to be high enough close to the surface of the ball to overcome that inwards gravity.

Well we can work this out in ballpark figures. (I'm not doing much checking so I may be out: you should check all these figures.) The following sort of "back of envelope" calculations is common in physics and serve as a reality check before we spend billions on supercolliders and spacecraft and so on. Hold on to your hat - here we go:

The density of air at std temp and pressure is about 1.25kg.m^-3 ... this is pure fantasy, there is no way our imagined ball of gas will have anything like this but we have to start somewhere.

1AU radius sphere of that air would mass 1.75x10³⁴kg - which is nearly 8900x the mass of the Sun... I make the acceleration due to gravity at the "surface" (1AU) of this sphere would be about 50m.s^-2 more than 5x that at the surface of the Earth.

Any proposed "container" will have to support itself against this (5g) pull - if you made the container out of Al-foil (13μm thick, density 2700kg.m^-3) you will need of order of 10²³kg of it - but this is easier of it is to be "inflated", which is what you are imagining - it would be like one of those foil balloons. I'm going to worry about the balloon's gravity later - you can see it is substantial.

To be contained by it's own gravity, the mean speed of the gas molecules close to the surface needs to be less than the escape velocity. Since this is only a mean, there should be enough to inflate the balloon.

I make the escape velocity at 3.95x10⁶m.s^-1 or about 1% of light-speed. This puts the temperature of the order of a staggering 10¹⁰K ... since the surface of the Sun is a scant 5700K and of the order of 10⁷K in the middle, I'm worried I got a number wrong so check this please.

What it means is, we are imagining a ball of gas that is very definitely contracting under it's own gravity. In fact, it's more like a protostar.

 

[writtenword]

In attempting to disconnect Gravity and Density I was just referring to the idea that the sphere need not be pressurized maximally. The amount of gas in the sphere could be more or less depending on how that affected things. But it is certainly still acted upon by gravity in all cases.

In some ways this model acts just like a pure equation in that we need not speculate on the nature of the sphere beyond its nature as a barrier. I did describe it as a structure but I am afraid that was merely a delusion of Dysonian grandeur. The sphere does not interact with the model other than containing it. Does that seem messy? I think of it as baby steps.

I resolved to put numbers to this until it cried out for fiction but that has come sooner than I had hoped. While I knew gravity was going to be a factor, I had not counted upon temperature. I was heartened by the estimation of only 5g at the shell which was very close to my own estimation. I put 1g somewhere at 30 million meters. Ah but then hope is dashed upon the rocks of an energetic reaction tied to compression. 10^10K is not so easy to handle reminding me that one should not dare math to a creativity contest, it has no sense of humor.

So what I have here is, at the very best, a proto-star and at the wost a cosmic cataclysm in a bag. To carve anything habitable from this I would have to leap immediately to magic or flip to a vacuum based interior with discrete bodies. I am certainly not above descending into the murky depths of “just making it up” but I had hoped to give interested parties something to figure if they felt so inclined.

To quote my Appalachian ancestors. “Well, dern.”. Still, I thank you for your aid and input Master Bridge. My left brain sheepishly agrees with your findings while my right brain insists that Ming the Merciless would totally know what to do at this point.

Perhaps I will move on to the atmospheric retention properties of irregular or oblate bodies.

 

[Simon Bridge]

Well... there is always making a smaller sphere, the size of the Superbowl? Australia?

Rotate your sphere + atmosphere so you can have mostly vacuum in the center and an atmosphere which varies in density along the surface from the "equator". You get a very flat Niven ring and a huge area which can be plated with solar panels or other energy collectors since you can have a Sun in the middle now.

There's a huge scope for interesting stories in this sort of structure.