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Does building a vessel such that it can maintain a breathable atmosphere in a vacuum (like all our manned space vessels) also makes this vessel watertight?

If not, why?

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At a first glance yes. If the inner crew compartments with breathable atmosphere are supposed to be air-tight to the pressure difference of at least one 1 atmosphere, and the hull is a rigid structure that's not meant to expand with change of pressure to within this 1 atmosphere, then it's fairly reasonable to assume the same pressure difference could be sustained by it in the opposite direction (where outside pressure is larger to the inner pressure for up to the same 1 atmosphere, or ~ 100 kPa, or ~ 14,7 psi, or the depth of ~ 10 m of water at sea-level Earth, where the Earth's atmosphere itself would equalize the pressure and the 10 m of water would increase external pressure to plus one atmosphere).

However, this is not necessarily the case, and vessel could be constructed to withstand a positive pressure (where inside pressure is bigger) better than the negative pressure (where the outside pressure is bigger), or even intentionally designed to expand with increased internal pressure, like some inflatable / expandable modules, e.g. Bigelow Aerospace's Genesis 1 and Genesis 2 expandable space habitat technology. These would clearly collapse under larger external than internal pressure.

Another concern with diving space vessels into the waters, if they're not designed for it, could be exposed electrical wiring, water soluble or water reactive materials used on its exterior, or pressure of water flow (waves, currents, vortices,...) damaging and/or bending equipment. And if their buoyancy wouldn't stabilize at the depth that it could still withstand external pressure, it could sink beyond its crash depth.

So your space faring vessel could experience problems due to the change of direction of the positive to negative pressure, and the properties of the water body, such as density, electrical conductivity, being a solvent to some materials, even facilitate corrosion and lose heat faster due to convective heat transfer. And the mass of this water your vessel would displace (buoyancy) would have to be roughly equal to the mass of your vessel, to prevent it sinking too deep and into external pressure it couldn't withstand any longer and implode. All of which are problems that are largely absent (or altogether different) than in the vacuum of outer space.

So in reality, no. Not unless they're specifically designed to land on water, like e.g. the crew capsule of the Soyuz spacecraft.

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Axiomatically, yes, an airtight craft is also watertight, since the molecules of water are less able to flow than those of air.

Current spacecraft are all rated for at least 1 bar (100 kPa) internal over external pressure difference. All are generally considered airtight (though all leak slightly).

A terminology clarification: by overpressure, I mean an internal pressure higher than the external pressure; by underpressure, I mean the internal pressure is lower than the external. In the differential I use + to mean greater internal and - to mean greater external.

There are, however, additional considerations. Direction of water resistance, direction of pressure, and duration of exposure all are relevant considerations.

Since spacecraft at surface generally are at +0 Bar relative (i.e.: 1 bar external and 1 bar internal) pressure, and at altitude are at +1 Bar, the bracing is all set to keep the pressure in. Certain fittings have in fact failed with less than -1 bar underpressure on apollo capsules. Note that water pressure rapidly climbs with depth in 1G - each 10m depth generates roughly 1 bar external pressure. Any Apollo, Gemini, or Mercury capsule was supposed to be able to withstand 0.5 bar under pressure for a short period during landing, and the capsule was held afloat by large balloons. And, with a singular exception, this worked. And the Apollo capsule flown was capable of ±0.8 Bar relative during pressure tests.

Further, many systems on the spacecraft are external to the pressure vessel, and while shielded from ram air pressures on launch, are not shielded from ambient static air pressure. Some of these are potentially damaged by water in quantity.

Pressure vessels designed for keeping water in have to deal with a mass thousands of times greater than those for air. So, an internal water environment capsule (say, for dolphins) would need to be braced differently for acceleration, and thus an air-design capsule cannot safely be considered watertight during launch, even if it would hold the full mass internally on the pad. Likewise, for movement through water, the forces are considerably higher.

One further consideration is that some of the seals are not viable for water environments (internal or external) except for very short durations due to chemical considerations.

So, practically, a spacecraft is not a viable submersible watercraft as an axiomatic condition, but is capable of preventing water intrusions within a reasonable range of exposures.

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