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I was reading in 'How could one navigate on Mars' about communications satellites, and it got me thinking about radio relay and navigation infrastructure. Pure nitrogen is an inert lifting gas in the carbon dioxide atmosphere there. Carbon monoxide is too and could be refined from the air. Hydrogen could be gotten from the water in the soil but would slowly leak from the balloon, though it is so much lighter and plentiful it would maybe still be a better choice. With the very thin atmosphere it would take very little to fill a balloon - but it would also mean a balloon would have to be extremely large to have much lifting capacity. Still, if all it is is an antenna for a radio transceiver it is tethered to, and a navigational aid, maybe it could be a good way to cheaply set up reliable radio contact on Mars and an array of landmarks for triangulation visible over great distances. Wind loads are a lot lower on Mars, there aren't any thunderstorms - though maybe static discharges would be a problem.

Could this be useful? How high would they need to be to get above most of the dust in a storm and thus work despite the static interference during such storms? What kind of calculations do you need to do to analyze such a system?

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During day they could be, but the main problem on Mars is the proverbial in troposphere during day, in stratosphere during night issue, due to extreme diurnal air temperature variations and air saturation swings between fully saturated during nights and extremely dry during days.

Height of dust storms is tough to appreciate with a few single numbers because of the so-called lee-wave clouds that rise and fall due to change in atmospheric pressure and local topography (elevations, depressions). For example, from NASA's Spacecraft Monitoring Martian Dust Storm article:

Starting on Nov. 16, the Mars Climate Sounder instrument on the Mars Reconnaissance Orbiter detected a warming of the atmosphere at about 16 miles (25 kilometers) above the storm. Since then, the atmosphere in the region has warmed by about 45 degrees Fahrenheit (25 degrees Celsius). This is due to the dust absorbing sunlight at that height, so it indicates the dust is being lofted well above the surface and the winds are starting to create a dust haze over a broad region.

So the atmospheric dust is widely dispersed in dozens of kilometers high layers, especially during the global dust storms, but also with more localized dust devils that can reach 20 kilometers high above ground:

   enter image description here

   A Monster Dust Devil Stalks the Martian Landscape (Image source: HiRISE, Credit: NASA/JPL/University of Arizona)

Fast winds are also not to be dismissed of, despite their seeming lack of force, because of the surface area of your lighter-than-air tethered balloons exposed to it. So it would have to be quite a system, though in my opinion not out of the question with on the cusp materials science. It would however at first likely have questionable reliability, or at least it would have to be established in-situ, so shouldn't be depended on for basic needs such as primary communications channels, or navigation markers / beacons.

Calculations would be more statistical and procedural in nature, from tracking atmospheric conditions over years, to evaluating the need for them and establishing reliability of such systems. But you'll first have to have exact designs for them, or even do trial runs in simulated environments.

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