The spiraling effect that I was talking about is achieved by arranging that a satellite's orbit is going to get repeatedly boosted by tidal effects. So you aim it so that it gets one boost. You then make a very small adjustment so that the orbit will get another boost. And another small adjustment so that the orbit will get another boost. And so on. The energy needed for these adjustments is far less than the size of the boosts, so you manage to move the satellite from one orbit to another with far less input of energy than flying there directly. But aiming a satellite so that it gets a series of those boosts without readjusting is impossible. Generally each one leaves you not quite lined up for the next. And even if it might, aiming it precisely enough to actually hit a sequence of boosts is an act of precision far beyond our capability. (I would not be surprised if it requires knowing position and momentum below the limit of the Heisenberg Uncertainty Principle.)

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.

Therefore this effect can be used to move a satellite to the space junk. (Albeit very slowly.) It can't be used to easily send the junk on a collision course to the Earth.

Now let's move on to economics. You're right that public goods can be provisioned in some special cases by public markets. As I pointed out in my first post (with examples), and as is discussed at great length in The Logic of Collective Action, public goods get provisioned by free markets when the benefits to some individual or small group of provisioning them outweigh the costs. (They also can get provisioned through organizations in a variety of ways, which inevitably involve some form of coercion. For details go read the book.)

Given the extremely low cost of distributing software, the cost threshold of providing software to the world is very low. It furthermore turns out that there is no shortage of potential returns for an individual that justify contributing. And the result is that the public good of open source software is provisioned amazingly well. All of this fits perfectly with the economic theory. In fact I remember noticing a passage in the book which describes this state of affairs and correctly says what would happen, but then goes on to say that it is extremely hard to imagine a public good where this would be true. (Do remember that this book was written back in the 1960s.)

However software is a very unusual case. And your experience with open source software should not mislead your intuition about the problem of space junk. That has far higher costs and far lower benefits to whoever deals with it, with very predictable consequences.

About the benefits of removing the junk. You claim that you've repeatedly pointed out that they are high (loss of satellite) and I've ignored you. That's a blatant lie and I'd like an apology. Please read Re^8: "Practices and Principles" to death and search for it is easier to just avoid stuff. Go to Re^10: "Practices and Principles" to death and search for 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. (The odds of any particular piece of junk and satellite having 2 potential collisions is ignorable.) Which is a capacity that all satellites have.

Let me emphasize the logical consequence. Since I've brought this point up multiple times, and its importance has not yet sunk in, I'm going to make it big and bold in the hope that this time it won't sail over your head.

No satellite operator will pay more to remove a specific piece of junk than the cost of maneuvering their satellite out of its way. Which is orders of magnitude less than the cost of removing that piece of junk by any technique known to man.

Meaning that nobody will ever choose to remove that junk. And, of course, if nobody ever removes any piece of junk, then all of the junk stays there until it is removed by natural causes. Which usually means atmospheric drag - a process that generally takes centuries.

A couple of things you are not considering:

1. 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.

2. 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.

3. 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.

4. 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?

5. 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.

"Too many [] have been sedated by an oppressive environment of political correctness and risk aversion."

I don't have energy to respond to this in full. However I am sad to tell you that you're mistaken on many counts. For instance contrary to your claim that I've implied that atmospheric drag begins at a low level, I've said that atmospheric drag will bring junk back to Earth in a time frame typically measured in centuries. (I just double-checked it and found I was wrong. While pieces from that may go for centuries, a lot of it will get fixed much faster.) For another example you got the relationship between size and orbital decay exactly backwards - larger pieces of junk experience more drag but have far more inertia and inertia wins. The same scaling principles that makes small things more affected by air down here operate in space. Another simple mistake is your claim that small particles are in streams. Consider the Chinese space satellite collision - that generated an estimated million pieces 1 mm or bigger and I guarantee it is not in a stream! Also even if you start out with a stream of small particles, miniscule velocity differences will, over the course of years, result in them being many km apart. And even if the initial speed was the same, differing effects of atmospheric drag would move particles apart.

Also there are major difficulties that you are minimizing. It is true that I assumed that you need to rendezvous with the junk to deal with it. It is true that you can more easily find collision courses with it. But read Re^7: "Practices and Principles" to death for how difficult it is to work with collisions at that speed, and recall that any shrapnel is new junk. I think my assumption that you want to match speeds holds!

This may be a good point to point out that a satellite in a Molniya orbit as you suggest using is going to be very far from still relative to any piece of debris it encounters that is not itself in a Molniya orbit. That is because while the satellite is fairly still relative to the Earth's surface, the piece of debris is nowhere near still relative to the Earth's surface, so there is a large relative velocity. In fact this is a general principle. If you encounter a piece of debris and at the point of encounter you do not have a large relative velocity, then you and the debris must be on very similar orbits! And conversely if you're on different orbits, any encounter will be at high velocity. The reason is simple, it is because from your position and velocity you can calculate every aspect of your orbit. So if your position is the same and your velocities are close if and only if you're on very similar orbits.

As for reusing existing spacecraft, review the link above about what collisions look like in orbit. Consider well that shrapnel is new junk. And then I think you'll agree with me that this is an approach that is more likely to create problems than solve them.

This is hardly an exhaustive list of issues I can come up with. (For example I didn't want to get into economic issues.) But it is enough to show how hard it is to solve the problem of space junk.