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Imagine a people who live within the debris ring of a planet. They make their homes on rocks that are only dozens of miles in diameter, so escape velocity is not a concern. They also do not need to travel very far: when orbital alignment is favorable, two rocks might be as close as 20,000 miles.

Unfortunately, the primary means of propulsion used by these people is compressed air. How big do these ships need to be to get from one rock to another?

Ideally, the journey should not take longer than 2 days.

I tried to work this out myself using the rocket equation but my smooth brain can't figure it out.

Specs

  • For the tanks of air, consider this commercial product. It seems to weigh about 700 lbs empty, has a capacity of 400 gallons and is rated for 165 psi. One way of answering "how big" this ship needs to be would be to indicate how many of these tanks would be needed to make a 20,000 mile journey in 2 day.

  • The payload is a cabin for two humans that weighs 2000 lbs.

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    $\begingroup$ This question may get better answers in world building stack exchange, to maximise the chance of getting answers useful to you suggest looking at solar sails and either using them, or specifically excluding them to get answers about cold gas rocket ISP. $\endgroup$ – GremlinWranger Dec 18 '20 at 22:47
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    $\begingroup$ What's important is not how far apart they are, but what the difference in their orbital distance is. They're likely to spend more propellant avoiding debris than performing the transfer. $\endgroup$ – user20636 Dec 18 '20 at 22:49
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    $\begingroup$ @GremlinWranger That stack can be great for some things, but they have a habit of ignoring the actual question and instead answering what they think the true question is. I very specifically want to know about traveling with compressed air: on WorldBuilding, I'd get a dozen answers suggesting alternative means of propulsion (like solar sails). $\endgroup$ – Pink Sweetener Dec 18 '20 at 23:30
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    $\begingroup$ Should fit well in the "Trajectory design, orbital and celestial mechanics" category, but could need some editing to make that clear. $\endgroup$ – SE - stop firing the good guys Dec 18 '20 at 23:50
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    $\begingroup$ @OrganicMarble Thank you for helping me improve my question. I have added some additional information. $\endgroup$ – Pink Sweetener Dec 19 '20 at 4:58
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Edited to remove a maths error in using the rocket equation

This is a cold gas thruster, so theoretical ISP of around 80 (breathing air being mostly nitrogen). Using the rocket equation. If we assume 10% of the craft mass is compressed air this gives 82ms budget for maneuvering, if we make 90% of the mass propellant this goes up to 1805ms. While simple to design and safe to this is why real world cold gas systems only show up where very low DV is acceptable, since it does not take much performance improvement from an ISP of 80 to justify extra equipment mass in the rocket hardware.

20000 miles in two days stops being orbital dynamics and becomes a brute force torch ship profile. A straight line trip of 20000 miles in two days needs a velocity of 833 miles and hour or 372 m/s (and we want to stop so twice that), our 10% gas craft only has 78.4 total ms velocity, so not good enough, going to 60% makes it possible for a one way trip.

In practice it is not the distance that matters but the velocity difference in orbits. For earth moving from 500 to 1000km up (LEO) by definition objects will regularly be less than 500km apart, but if they hohmann transfer they will do the maneuver closer to opposition and travel half an orbit to close the distance, but do so efficiently. Looking at something like tells us a total DV of 260ms and around an hour. This is beyond the capability of a 10% by mass craft but a 30% one could do it. if out around lunar orbit (not not near the a moon) at 300,00km then a 500km orbit change involves less than a meter of DV, and a 12 day trip so very possible with pretty much any thrust system, as long as the 12 day timeline was acceptable. If more fuel available then the time can be reduced, potentially back to the straight line approximation where two 100ms 'burns' (25% gas by weight) do the trip in around an hour and a half.

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  • $\begingroup$ Thank you for your very detailed answer! I'm not sure why you are insisting on a craft that is only 10% gas though. It is my impression that most spacecraft are 80%+ fuel. Ion drives are beyond the scope, but thank you for the suggestion. $\endgroup$ – Pink Sweetener Dec 18 '20 at 23:35
  • $\begingroup$ This covers all the important points, but the "95% gas" part does not appear to be quite right. An Isp of 80s and a mass ratio of 20:1 is an overkill delta-v budget of 2350m/s. The mass ratio for a 372m/s budget is a much more reasonable 1.6:1, less than half the mass of the ship being propellant. $\endgroup$ – SE - stop firing the good guys Dec 18 '20 at 23:36
  • $\begingroup$ @SE-stopfiringthegoodguys yep, I've badly goofed the mass ration conversion. $\endgroup$ – GremlinWranger Dec 18 '20 at 23:42
  • $\begingroup$ Then I can safely go to sleep trusting everything will be in order when I wake up. $\endgroup$ – SE - stop firing the good guys Dec 18 '20 at 23:46
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    $\begingroup$ @PinkSweetener spacecraft that are 80% propellant don't carry their main propellant in the form of a gas. Gases have very low density, increasing that density involves increasing their pressure and using heavier tanks to store them. Outside of certain special cases, it's much more practical to store liquid propellants. $\endgroup$ – Christopher James Huff Dec 19 '20 at 1:04

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