Spaced Nerdism

The science bug done bit me again. I saw someone say that the way Rocky busted out of his enclosure to save Grace is a narratively nice change in the movie, but a bit dumb because the ammonia would definitely kill him. I wanted to know if that would be true. I used to do this kinda thing back in high school with fun little factoids spouted in Star Trek, I'm kinda curious to see if I can still do it. Also, it's useful for me and my fic to know if the parts-per-million of ammonia would be enough to cause lung damage, and also enough to justify some theories I have about the oxygen mask that Armando puts on Ryland after Rocky saves him.

I want to preface this with a few things.

I am a only an expert in the medical stuff, the chemistry and physics, not so much. I'll try to muddle through but expect errors.

I'm doing this by googling what kind of formulas I would need to find this stuff out, so again, please tell me if I'm using the wrong thing and I'll fix it!

I'm not above assuming best-case scenario numbers, Grace and Rocky already appear to be the luckiest bastards alive to have survived the entire movie.

So the volume of the Hail Mary is difficult/impossible to know precisely but it's said to be 4m diameter (what they could launch into space at the time). That 4m diameter is in the book, but I do believe in blending both canons. I'm also only doing 3 decimal places cause I don't hate myself.

For this thought experiment this diagram of the main living segment of the hail mary is incredibly helpful. In the Adrian fishing scene, we can see that Rocky has an enclosure in the cockpit, and he gets into it via his ball (we see the ball detach and roll away) and we know this little enclosure doesn't extend back any further than like...the ground level of the cockpit because there's no enclosure in the mental health bay(Don't Go Crazy room), and his ball attaches at the bottom of the cockpit, so it appears that's as big as that segment is.

I was able to scale the image on my screen to match the key there in the middle, which is 100ft long (or 30.48m which is what I used, I matched it to 30.48cm). The cockpit is 7.3m long. The corridor it attaches to is also 7.3m long before it branches into the EVA room.

Volume of a cylinder is π*r^2*h, making the volume of the cockpit 91.73m^3

So Rocky's enclosure is about a quarter of that volume, making it 22.9m^3. His atmosphere is described as mostly ammonia with trace amounts of other gasses. We could estimate that trace amounts might total up to 10%. 90% ammonia then. And it exists at 210ºC (483.15K) and 29atm (2938425 Pa). From the book, the Hail Mary's internal pressure is kept at 0.4atm (40530 Pa).

When Rocky breaks out of his enclosure he busts open a small area big enough for him to slip through, he looks to be about 50cm at the very widest? So maybe we assume a triangular space big enough to fit something 60cm opened up to let him out. An equilateral triangle with a midsegment of 60cm has a side length of 120cm.

Using my stopwatch I timed how long the area was open for when he was bursting out of it, and it was open 0.28 seconds.

Key:

n= amount of gas in mols m= mass in kg ṁ= mass flow rate (kg/s) P1= higher pressure (Pa) (2938425 Pa) P2 = lower pressure (Pa) (40530 Pa) V= volume of space (m^3) (Vr = 22.9, Vg = 91.73 T= temperature (ºK) (Tr for Rocky is 483.15, Tg for Grace is 293.15) R= gas constant, 8.314 in general, 488 for NH3 in Rocky's conditions, 259.8 for O2 in grace's conditions) Cd = discharge coefficient, I have no concrete way of knowing this but looking at what this is for trileaflet heart valves which are similar in principle, it's between 0.6 and 0.85 so we'll go with 0.6 because it's a sharp-edged and irregular opening. A = area of opening (cm^2) (for the equilateral triangle with 120cm sides this is 6235.38cm^2.) NH3 molar mass = 17.03g/mol. O2 molar mass = 31.998

Firstly we need to calculate how much ammonia is in Rocky's enclosure. We can do this with the ideal gas law. m = PV/RT = (2938425 * 22.9)/(488*483.15 = 67,289,932.5/235777.2 =285.4kg total in Rocky's enclosure. (90% being ammonia) =256.86kg of straight ammonia. We also need to know how much O2 is in Grace's immediate area. The book says his air is entirely oxygen at 0.4atm of pressure. Interestingly the scene where Rocky pressurises the tunnel and Grace is blown back by the door opening cause he didn't know and depressurised the airlock, the reading on his EVA suit says 21.1 kPa or 0.2 of an atmosphere, that seems quite low tbh. While it does equate to the oxygen percentage of air at sea level, it doesn't account for partial pressures of CO2 in ambiant air before it gets scrubbed by life support systems. Especially in the tunnel that Rocky made, since there's no airflow. Even with the door to the hail mary open, the circulation of air wouldn't be ideal with no fans directly within the space. NASA tends to require higher kPa of pure oxygen atmospheres to allow for human consumption and to reduce the partial pressure of expired CO2. So all of this to say, I've gone with the book-canon 0.4atm of pressure.

Given that the area Rocky busts out of is very close to the ground and the entrance to the corridor, I've counted the volume as being the volume for both the cockpit and the small bit of corridor connecting to it, which fun fact, they're the same length and diameter so it's just double the volume of the cockpit. m = PV/RT = (40530 * 183.46) / (259.8 * 293.15) = 7,435,633.8 / 76,160.37 =97.63kg of O2

Okay, so first to know which formula to use for the mass flow rate, we need to know if the flow of gas is choked or not, meaning we need to know if it caps at the speed of sound. To find that out, we need to calculate the heat capacity ratio of the ammonia, this is symbolised by γ.

γ=Cp/Cv where Cp is the constant pressure, and Cv is the constant volume. For Tr, the Cp of ammonia is 42.03.

Cv = Cp - R = 42.03-8.314 = 33.716

so γ= 42.03/33.716 = 1.246

So is the flow choked? if the pressure difference is less than or equal to the critical pressure ratio, then yes. It's a nightmare to type that equation out, so I did it on my whiteboard, and the result is 0.013 ≤ 0.559, which is true, and therefore the flow is choked.

So now we need to use the choked mass flow formula.

ṁ = Cd*A*P1*√((γ/R*Tr)*((2/γ+1)^γ+1/γ-1)) {good lord this is easier to write on a whiteboard) I decided to do this in two halves. first half works out to be: 0.6*0.624*2,938,425 = 1,100,146.32 The second half works out to be: (γ/R*Tr) = 1.246/(488*483.15) = 0.00000528 (2/γ+1) = 2/2.246 = 0.89 γ+1/γ-1 = 2.246/0.246 = 9.13 √(0.00000528 * (0.89^9.13)) = √(0.00000528 * 0.345) =√0.00000182 = 0.00134 Putting these two halves together we get: ṁ = 1,100,146.32 * 0.00134 = 1474.2kg/s ammonia released. The opening being open for 0.28 seconds we get 412.776kg of ammonia released. This is greater than the volume of ammonia even present, so the entire content of ammonia within Rocky's enclosure gets vented in this time.

ppmv = (mNH3/0.01703)/((mNH3/0.01703)+(mO2/0.031998)) x 1,000,000 =15,082.795/(15,082.795 + 3,051.128) =15,082.795/18,133.923 =0.831*1,000,000 =831,000ppm of ammonia.

I found this paper from StatPearls (Padappayil & Borger, 2023) that gives the levels of ammonia that will cause increasing levels of harm to the human body. It says, "concentrations above 5000 ppm usually produce rapid respiratory arrest" and "concentrations above 10000 ppm is sufficient to evoke skin damage". So he is for sure dying and all exposed skin is getting crazy chemical burns. Here is the paper for any who are interested:

I realised just now (after I've completed all the math and am typing this up), that the diameter is also measurable on the diagram. It measures out at 2.8m, but I've decided to leave it. As it stands the ppm of ammonia is already gonna kill him, making the space smaller will only make him more dead, and Rocky's enclosure will scale smaller with it, so for all I know the ppm will probably stay roughly the same. Onto Grace's resulting Injury!

Let's say for the sake of plot armour and Ryland Grace being the luckiest bastard alive, that the Hail Mary's life support systems are extremely overpowered and manage to scrub the air fast enough that he doesn't immediately die a horrifically gruesome death. Let's also say that it reduces it enough that he doesn't lose all of skin and go blind and go into respiratory arrest in the time it takes Rocky to get him to the medical bay.

The pathophysiology in the above paper outlines that inhaled ammonia readily reacts with water in human tissues producing an exothermic reaction that causes thermal injuries, or burns, resulting in chemical pneumonitis and non-cardiogenic acute pulmonary oedema (APO) with possible acute respiratory distress syndrom (ARDS), and lower airway obstruction from debris caused by epithelial and red blood cells.

APO is the accumulation of fluid in the airspaces in the lungs. The treatment for this is generally CPAP, or continuous positive airway pressure. This overall increases the pressure within the lungs to force the fluid back into the capilaries that supply blood to the lungs. It also helps to keep the alveoli (little air sacs) open, when the water changes the surface tension and causes the lung tissue to collapse with each exhale. The mask that is applied looks odd for just an oxygen mask. While the securement is just a little elastic strap, the amount of cushioning around it suggests they want a good seal, the ventilation tubing makes it more reminiscent of a CPAP set-up.

The injection is likely largely for effect, injecting directly in the internal jugular is a somewhat odd choice but very 'movie-drama' thematically appropriate. And given that his arms are rather inaccessible while he's in the EVA suit, I suppose it does kind of make sense that Armando would go there.

The medication I can think of that might help the most here is some kind of corticosteroid to manage the inflammation in his lungs. Second best guess is adrenaline, if he were truly peri-arrest that would help, and it'd also help with brochospasm and airway oedema. It's not likely to be a sedative since they're just using a non-invasive ventilation set-up and he needs to maintain his own airway. It's also possible but not likely to be a diuretic since the APO is non-cardiogenic.

He'd also likely be given inhaled brochodilators/further corticosteroids/nebulised adrenaline through the CPAP system. Long term he'd have chronic lung disease at the very least, namely bronchiectasis and pulmonary fibrosis. Bronchiectasis is the scarring and permanent widening of the airways, impairing the body's ability to clear mucous, creating a breeding ground for bacteria. This causes a predisposition for infections, chronic cough, fatigue, shortness of breath and wheezing, and haemoptysis. Pulmonary fibrosis is similar, it is the scarring and thickening of the alveoli. It also causes shortness of breath, persistent cough, fatigue, and weight loss.

So yeah, that's the result of the thought experiment! Also, here's the whiteboards for proof that I did the math, and also much easier reading than typed out formulas.