We're specifically referring to small changes here to service satellites in similar orbits.
Also, for future reference, please consider this site's guideline "Please respond to the strongest plausible interpretation of what someone says, not a weaker one that's easier to criticize. Assume good faith."
Nothing about your post confined the discussion to "similar" orbits; you were in fact responding to a parent describing a need for these to be precisely matched with a negation. While you're correct that there are orbital changes that are cheap, there are many that are not as well, and your statement without that qualification seemed likely to mislead.
I wasn't assuming bad faith on your part; I merely believed that your response could be confusing. You might consider that guideline as well.
The context of the conversation was that OP's comment was questioning the utility of a depot since they believed spacecraft in similar orbits would not be able to access it, only those that happened to be in precisely aligned orbits could. Interpreting "spacecraft can change their orbits" to be a claim that there are no limitations on the maneuvers a spacecraft can perform is very unreasonable.
the "for such an orbit change" is the qualifying statement there, as opposed to "for any orbit change". It is clearly referring to the same category of orbit change being discussed in the preceding sentence.
Plane changes can exceed the cost of putting a satellite in orbit, and even if they don’t they’re often more expensive than spacecraft can afford. The formula for changing a circular orbit without changing the altitude is 2V * sin(delta degrees / 2). This results in some pretty surprisingly high numbers; a 45 degree plane change costs 76% of your current delta v, a 90 degree change is 141%. Even a measly 5 degree change should cost 8% of the craft’s current velocity.
8% doesn’t sound like a lot, until you start doing the math on how much that is compared to orbital velocity. The Space Shuttle’s Orbital Maneuvering System was good for 300 m/s of delta v, which is only 3.8% of the minimum speed to keep orbit. If the shuttle was flying faster, those numbers start to fall.
I think at best the shuttle could probably afford to change its orbital plane by maybe 2 degrees.
I always assumed changes to altitude rather than plane for servicing orbits. Not an expert in this. I guess it depends on where high value orbits would be concentrated and space is big unfortunately.
Anyways, it'll be interesting to see where this all develops. Starship is clearly going to do in orbit refueling for their moon and other missions so there are clear situations where the delta-v for in orbit isn't too bad - match the plane on launch etc.
Let's remember that a 5 degree change in plane shifts the satellite over 590 km. If we pretend for a moment that satellites in LEO were evenly distributed, that 5 degree slice would contain the orbits of 53 of them. In reality, satellites tend to be placed in similar orbits.
Also your equation is for circular orbit inclination change at constant altitude, in practice you can do much more efficient plane change maneuvers by going into a higher and more elliptical orbit, changing inclination, and then returning to your original altitude and eccentricity.
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