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There's an article published by NASA in which they discuss the practicality of using an evacuated blimp or airship on Mars. Their main points are that since the exterior pressure is so much lower, this could be constructed with a lattice of existing materials in a large sphere. They also predict that:

Through a more in-depth analysis of the vacuum airship model, it can be shown that the vacuum airship may theoretically carry more than twice as much payload as a modeled dirigible of the same size, a 40-meter radius, in the Martian atmosphere.

However it seems to me like the numbers don't add up...

If 1 cubic meter of atmosphere on Earth has a mass of about of about 1.225 kg/m^3 (asl) and the atmosphere on Mars (asl) is 0.6 percent of that then every cubic meter of vacuum can only displace 0.00735kg of Martian atmosphere. Now sure, Mars' gravity is also less but I'm still not quite sure how someone would construct a solid structure for less than 0.018kg which can contain a vacuum, even in mars' low pressure environment. Am I missing something? Where did my assumptions go wrong?

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    $\begingroup$ I think the comparison is to a dirigible on Mars (helium-filled), not on Earth. $\endgroup$ – SF. Nov 15 '17 at 15:08
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    $\begingroup$ @SF. Oh I see what it means. That answers one of my questions, thanks! $\endgroup$ – Dragongeek Nov 15 '17 at 15:33
  • $\begingroup$ Just going by buoyancy, hydrogen/helium are nearly the same performance as vacuum. I assume the quote about the payload comes from analyzing the overall mission architecture. Most notably, a standard blimp would need to carry along spare gas to account for losses over the course of the mission which could add up fast. $\endgroup$ – Lex Nov 15 '17 at 15:52
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    $\begingroup$ Your error is just to multiply 1.225 with 0.6 %. You have to consider the different gas mixture of the atmospheres of Earth and Mars. On Mars there is mainly carbon dioxide, on Earth mainly nitrogen. The density of CO2 is higher than that of air. The lower temperature of Mars increases the density. $\endgroup$ – Uwe Nov 16 '17 at 17:42
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For the sake of this answer I'm going to use the numbers I get from NASA and Wolfram Alpha. These are:

Density of Earth Atmosphere: 1.204 kg/m^3
Density of Martian Atmosphere: .020 kg/m^3
Density of Helium on Mars: .001458 kg/m^3
Density of Helium on Earth: .1663 kg/m^3

Volume of a 40m radius sphere: 268083 m^3

These assume 1 atm pressure and 20 degrees C on earth and 6.36 mbar and 210 degrees K on Mars (sorry for the unit mix up).

That means that the Helium craft displaces:

Earth: (1.204 - .1663) * 268083 = 278185 kg
Mars:  (.020 - .001458) * 268083 = 4971 kg

And the Vacuum craft displaces:

Earth: 1.204 * 268083 = 322772 kg
Mars:  .020 * 268083 = 5362 kg

These numbers will change dramatically depending on the temperature, which can vary widely on Mars, so even though I didn't get the same result that Nasa did, it's entirely possible they're are assuming a colder than average temperature.

Something I didn't note earlier but I think is actually kind of important: the density of a vacuum doesn't change (by definition), so the weight of the craft will not vary with temperature, while the helium craft will have to add or remove helium to maintain a stable pressure. This means that it will perform much better as it gets colder in comparison to the helium craft, which becomes more dense as the temperature decreases.

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    $\begingroup$ The NASA study may actually be contrasting with a less potent lifting gas given that hydrogen and helium could be difficult to transport to the Martian surface in large quantities. $\endgroup$ – Lex Nov 15 '17 at 16:30
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    $\begingroup$ Hydrogen especially is known to leak? seep? through the solid walls of fuel tanks. $\endgroup$ – Baldrickk Nov 15 '17 at 17:08
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    $\begingroup$ To the anonymous enditor attempting to add material to this answer: editing in additional material would be better done with the consent of the author via a comment, hence my rejection vote. $\endgroup$ – Puffin Dec 29 '20 at 17:20
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    $\begingroup$ @Lex At Mars you can make hydrogen by electrolyzing water. Helium would be harder to come by, and leaks through seemingly-solid materials even more than hydrogen. On Earth helium comes from natural gas wells, as a minor contaminant in the natural gas. Radioactive decay in Earth's interior produces alpha particles that swipe electrons and become neutral helium atoms, that slowly make their way into the near-surface crust. The same would happen at Mars, but without the natural-gas infrastructure to harvest it. $\endgroup$ – Tom Spilker Jan 5 at 23:25
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    $\begingroup$ One difference between a standard balloon and a "vacuum balloon" is the stress state of the envelope materials. A standard balloon can have the entirety of the envelope in tension. A vacuum balloon, essentially a big vacuum chamber, is necessarily in compression, and compression members can buckle, which tension members cannot. The structure to support the compressive loads is heavier, per contained volume, than one supporting tensile loads, so you lose some of the advantage of having no contained gas, which was not that significant anyway. $\endgroup$ – Tom Spilker Jan 5 at 23:30

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