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This answer to Is there an upper limit for the internal size of space stations? has this paragraph:

Finally, if you are stuck ["stranded" in the middle of a room in a space station filled with air], you are no longer affected by any decay in the space station orbit. The ISS changes speed, and thus altitude, because of the slight amount of air friction, and must be periodically boosted back up. Eventually the space station will come to meet you. The time scale for rescue would depend on the state of the earth's atmosphere and the shape and orientation of the station.

It got me wondering which way the space station would move relative to the floating astronaut inside it as direct effect of the drag, without anything getting "boosted back up".

It seems to me that the space station will in fact accelerate mainly forwards and slightly downwards at the same time, as it goes into a lower, faster orbit. But the cause is air drag which is a force acting backwards, and because of F = ma that means an acceleration backwards at least initially (I think). But for how long? An infinitesimal time? It seems to me that what makes it hard to think about is the continuous action of the drag.

My question is: What is the acceleration in an orbit decaying due to air resistance relative to one not decaying?

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    $\begingroup$ Relevant: space.stackexchange.com/questions/12631/what-is-the-iss-drag says the acceleration is 0.656micrometers/s² at orbital altitude of 400km. But depends on stations attitude, space weather, solar sail alignment ... $\endgroup$ – user40414 May 1 at 20:35
  • $\begingroup$ The acceleration due to air resistance will accelerate the station backwards. The lower the station gets the greater the air resistance and the greater the deceleration $\endgroup$ – Slarty May 1 at 22:11
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    $\begingroup$ @Earthworm that's a great link! Readers should note that the variation of acceleration with solar activity covers a huge range, more than a factor of 10. There will also be day/night variations as well. How steady is the atmospheric drag force experienced by the ISS? $\endgroup$ – uhoh May 2 at 1:00
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    $\begingroup$ @MatthewChristopherBartsh I added the title of the linked question for context, which is the possibility of getting stuck near the center of an extremely large hypothetical space station, not something like the ISS. If the directional acceleration is too low, the astronaut could get caught in an airflow whirlpool perhaps, slowly dehydrating until rescue or death. See Tiny emergency propulsive device if stuck floating in a large volume in microgravity $\endgroup$ – uhoh May 2 at 1:01
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    $\begingroup$ @uhoh An airflow whirlpool would be about the only thing that could cause anyone to get trapped in the middle of an air filled room. I hadn't thought of that. I wonder how such a whirlpool would work? $\endgroup$ – Matthew Christopher Bartsh May 2 at 2:41
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But the cause is air drag which is a force acting backwards, and because of F = ma that means an acceleration backwards at least initially. But for how long? An infinitesimal time? What makes it hard is the continuous action of the drag.

Drag for the orbiting station is has the effect of a continuous rearward force acting on the station but not directly on the astronaut. So the station is constantly accelerating backwards relative to its contents.

This acceleration is minuscule, and not going to be noticed by the astronaut inside, of course.

It seems to me that the space station will in fact accelerate mainly forwards and slightly downwards at the same time, as it goes into a lower, faster orbit

The acceleration that makes the station go into a lower faster orbit is due to gravity, and gravity acts on both the astronaut and the station nearly equally. There's a small tidal force as the astronaut moves away from the center of mass of the station, tending to move the astronaut higher or lower in their orbit, which is of similar magnitude to the drag term.

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