A couple of things you are not considering:
Junk is individually easy to dodge. That makes this the third post in a row where I've pointed out that the real cost of any particular piece of space junk to any existing satellite is that the satellite might have to make a maneuver. Once. plus other similar statements.
Have you read about the Kessler Syndrome? Tommaso Sgobba, Director of the International Association for the Advancement of Space Safety is to present his case to the United Nations in April.
Also, the 9,000 pieces of space junk we are currently discussing are just the tip of the iceberg. Those over 20 cm in size that are being tracked. There are an (estimated) 4,000,000 lbs of junk inclusing 110,000 over 1 cm in size. And a speck of paint was capable of digging a 1/4 inch wide pit in a shuttle window. ref.
Commercial satellites (like the geosynchronous Astra STV Fleet of 15) have to constantly adjust their positions in order to maintain their footprints. There manouvering fuel is strictly limited (weight == cost * 5 for anything destined for geosynchronous orbits). And that fuel is a major constraint upon their service lives. The amount of thruster fuel taken into orbit is carefully calculated to allow these maintenance manouvers over the projected lifetime of the satellite. Using that precious and expensive cargo for debris avoidance is a direct hit to the bottom line.
In addition to that, a piece of junk in geosynchronous orbit has to be as big as a basketball for it to be tracked by radar, so there is a lot of debris that could damage a satellite that simply cannot be avoided because we don't know its coming. That leaves the problem of how would we know where to go to deflect it, but that's a different question.
- Your posts to date imply if not state that atmospheric drag only begins at some low level (100 miles or 100 kilometers).
Atmospheric drag is present throughout the LEO orbit range. Right up through 2000k, to some measurable extent. It increases the lower you are obviously. To wit: During the last shuttle mission, the shuttle's thrusters were used to push the ISS higher to compensate for accumulated drag over it's life to date. It could have done this using its own thrusters or those of the attached Progress vehicle, but as the shuttle was about to return and had spare fuel, that was used instead.
So, if you can deflect a piece debris lower, you increase its orbital decay component. The lower you knock it, the greater (exponentially) the effect. You don't have to push it all the way to the 100k/100m boundary in a single go as with the Hohmann Transfer orbit. (That's what I meant by one-shot!).
If you deflect its orbit so that it perigee is half way there, the increased drag will slow it further and the perigee on the next orbit will be lower still. Hence more drag, further slowing, lower perigee until burn up. (Ie. In the long term, all LEO orbits are spirals!)
How far down do you have to knock it in order to appreciably shorten it life? That's where the math goes beyond single formula posted on wikipedia. The bigger the debris, the greater the drag. So smaller items have to be kicked lower to achieve the same end date. But, the smaller (lower the mass) of the object, the less kinetic energy it has, so you can (perhaps) use a greater deflection angle for the collision.
Also I note, again, that if you get your garbage satellite to the junk, it is much simpler to just give your satellite a garbage can, and put the junk into it. The real cost here is the effort of maneuvering your garbage satellite.
Your assumption is that you have to exactly match the debris' orbit in order to deflect it. You certainly would to pick it up.
But: if you look at the Molniya orbit, you will find that it is possible to achieve a similar affect to a geosynchronous orbit at a much lower altitude. Ie. a satellite can 'park' over a particular spot on the earth surface for a protracted period of the day.
If you consider that parking spot above the earth's surface as the piece of debris that you wish to rendezvous with, then the satellite and that spot are, for a period of 12 hours or more, moving such that they have a relative angular velocity of 0. And, in the case of the Molniya orbit, the angle of the orbit is chosen specifically to minimise the perigee shift.
So now imagine that the deflector is in a similarly highly elliptical orbit carefully chosen to rendezvous with the chosen piece of debris (and it's circular-ish) orbit for a briefly co-incident period in which their relative velocities was low enough that the collision is gentle enough that the debris did not fragment. This matching of relative velocities only has to be for the instant of contact.
Further more, the two orbits do not have to even be in the same plane if you rotate the deflector to be at right angles to the path of the debris.
Indeed, there may be some benefit in presenting the deflector at an angle to the path of the debris in both the vertical and horizontal component. Not only does this further enhance the "glancing blow" effect. It can also be used to transfer debris flying in polar-ish orbits into equatorial-ish orbits where the atmospheric drag extends out further.
Then again, maybe the thing to do is not actually attempt to cause the debris to burn up. Maybe it is better (more feasible) to simply push all the debris into a well-defined parking orbit.
Another notion is that, for the very small particular debris, which I believe tends to orbit in long streams rather than as individual pieces, you might use an aerogel swat--similar to the way they captured the comet particles on the Stardust mission. The reverse side of the titanium plate could be covered with aerogel and presented for appropriate encounters.
- You've also been (implicitly) assuming that you have to manouever the deflector into a specific orbit for each piece of junk.
Imagine putting the deflector into an orbit such that it (serially) becomes co-incident with 2 pieces of junk. a few minutes or hours or days apart. Hey! Two for the price of one.
Now consider building a database for the 9000 pieces of debris (less because some are just too large to consider moving), that plots their positions at say 1 second intervals over the next month or two. Now you search through that dataset looking for multiple pieces of debris that can be rendezvoused with using a single orbital path conducive with the other constraints and requirements. Remembering that you do not have to match their speed and orbit, only intersect with it at the right time at the right relative velocities. With LEO orbit taking ~90 minutes, it's not hard to see that several connects per day or week or month might be possible. With the ability to alter the angles of the deflector between rendezvous for little cost to account for disparate angle of approach.
Sure it's an NP hard problem, but the N is sufficiently low to make it tractable. It's far less hard than fluid dynamics problems and your average F1/Kart racing team has the processing power to take these on. With NASA/ESA/JAXA tackling the problem?
- You also mentioned (elsewhere) the cost of building all these junk collector satellites (paraphrase).
One of the key elements of the idea was that we would be re-using a shuttle that already exists and simply adapting existing technology, due to be scrapped, for this purpose. Yes there would be launch and adaption costs. And you have to consider what you do with the thing at the end of its useful junk management life.
Maybe a consortium of free enterprise guys could meet the upfront costs and earn money from the missions? Or, maybe those minimal costs could be shared by the major space agencies.
The whole of the non-commercial space endeavour is currently publicly funded in the name of science. I see no particular reason, if the idea was feasible and the benefits demonstrable, that the four major players wouldn't stump up say, $10 million each in order to put the vehicle up there. In the name of science and mutual benefit.
And at the end-of-life, as the shuttle wouldn't need to return to earth. The main engine fuel usually carried for de-orbit burn can be used for boosting the shuttle into a graveyard orbit.
I'm avoiding any real depth of discussion regarding the financing and merits of public versus private enterprise--though I have my opinions--because that would be entirely pointless discussion if the notion is infeasible on theoretical grounds. So far, nothing you have said yet convinces me that it isn't. Or is.
I guess what I am saying here is that I'm not ready to abandon thinking about the notion on the basis of your extrapolations from simple case, single encounter calculations. Indeed, they have just stimulated my thoughts and research further. For which I thank you. And curse you :)
Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
"Science is about questioning the status quo. Questioning authority".
In the absence of evidence, opinion is indistinguishable from prejudice.
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