Uvazhayemiy Gospodin Koptev! ("Respected Mr. Koptev!")

Regrettably, it is now too late to save your space station Mir for posterity. As an alternative to the wasteful thalassosynchronous disposal orbit you are contemplating, consider this modest proposal instead. After thoughtful analysis, I think you will agree that it offers a far higher rate of return than your present plan.

Let us pretend for the moment that we are insurance underwriters reviewing an application for covering the liability due to an uncontrolled reentry by your space station Mir. Automobile collision policies are usually broken down into bodily injury and property damage, so let's use those terms.

In its 51.6 ° orbit, Mir overflies 400 million km ² of potential Ground Zeros. About two thirds of that is water; essentially all of Earth's 6.2 billion ¹ people reside in the remaining third, so P_land = 0.33

Because they are unpowered, the various bits of Mir which survive reentry will follow ballistic trajectories - so by sea level they should be falling nearly straight. An average human being standing in the open would present a target area of 0.2 m ² from above (0.6 m wide x 0.3 m thick). Multiplying the individual target area times number of targets (people), and dividing by land area, yields the probability that any particular impact site will be occupied. P_{warm body} = 0.000008, or roughly 125,000:1.

In all of recorded history, no one has ever been killed by a meteorite, let alone a satellite, but one American lady was badly bruised on her hip by a fist-sized rock back in the late 1940s or 1950s. So let's take that dimension, say 10 cm, as the lower limit of lethality, and approximate stepwise for simplicity's sake. Assume that debris which isn't vaporized in the upper atmosphere will be distributed 50/50, i.e. half above the limit, and half below. Furthermore, let's stipulate that projectiles under the limit will do nothing, while those above the limit will kill the target. So P_lethal = 0.5

It has been estimated that as much as 40 metric tonnes of robust metallic parts might survive reentry. Assume what survives is basically solid and roughly spheroidal. 40 tonnes of stuff at an average specific density of 3 (solid aluminum is 2.75, steel is 7.8, a typical stony meteorite is 3.4) makes 13 cubic meters, which if formed into spheroids averaging 10 cm across, makes 26,000 potential impactors. Lightweight stuff probably won't survive reentry or be going very fast when it hits the ground. We can be sure that the material will be spread out all over the place, so each object will have an equal chance of hitting someone, and that lightning probably won't strike twice, so to speak. So N = 26,000.

Bodily Injury: Multiplying

yields a 1:370,000 chance that any given chunk of Mir will whack at least one citizen (voting or not). But there may be 26,000 such chunks, half of which could be deadly. The joint probability of kill can be expressed as:

The chance of no one being killed, P_safe, is simply the probability that every individual attack will fail, i.e. 1 - P_{kill, j} = 0.966, or 32:1. While these events are not truly independent, and Mir is not falling all over the world, just a small portion of it with considerable clustering, this should set an upper limit on the risk. The record jury award for wrongful death in the USA is US$108 million. Insurance actuaries use figures 10-30 times lower. I also note that nations such as Egypt have been renting landmine clearing contingents to the UN for as little as US$3 per day per man (industrial lifetime in a minefield ~2 months), which certainly sets a lower monetary limit on human life. Given that the rest of the world outnumbers litigious Americans by 22:1, the expected bodily injury liability from an uncontrolled reentry should fall in the range US$0 - US$3.7 million.

Property Damage: It has been estimated that the value of the physical plant (homes, factories, infrastructure, personal property, etc. but neither land or money) belonging to the United States can be replaced in 4-7 years of total effort. Furthermore, the US share of the global product is about 28% ². Thus, a quick-and-dirty estimate of the total value of global physical wealth excluding currency and land is: $10 trillion x 7 years ÷ 0.28 = US$250 trillion. Since little of that wealth is in the polar regions, let's assume Mir flies over all of it. The specific value works out to 63 cents per meter². The atmosphere will dissipate most of the kinetic energy, and incinerate any toxic chemicals, plastics, etc. Freely falling, unfueled, unpowered Mir parts should be limited to a terminal velocity about Mach 1 near sea level. Although impact energy is a cubic function of radius, the range of a given destructive effect scales linearly with radius. Assume that direct blast effects from the 13000 impactors above the lethal size limit extend 10 radii from Ground Zero, while the 13000 objects smaller than that damage only what they actually hit. The expected value of property damage is [pi x ((0.50² x 13000) + (0.05² x 13000)] x $0.63 = US$6500, about a tenth of one percent of the upper limit on bodily injury. So PD can be ignored.

Recognizing the large uncertainty in these estimates, it seems that the expenditure of a $14 million Progress flight to control this event was a prudent investment. As a precedent, I note that your predecessors in the Soviet Union did eventually pay out about US$3 million for the RORSAT's nuclear reactor which scattered itself across upper Canada in 1978.

However, since scarce resources have already been spent to avoid this debacle, why not maximize the value? You are surely aware that genuine Russian space souvenirs have fetched very high prices at auction, as did much rarer Skylab debris. Speaking for myself, I would certainly pay some nominal amount, say $9.95 including shipping and handling, for a gram of genuine Mir encased in a lucite cube on my desk. 40 tonnes times 10 bucks per gram makes US$400 million gross value; even more if your Mission Control takes steps to maximize the survival and collection rate of the reentering debris. Moreover, this figure doesn't include what extreme tourists might be willing to pay to witness this event first hand, or what the media would pay for real time footage.

The best way to maximize souvenir collection is not to drop them in the drink. It is a lot easier to pick hot smoking bits off the ground than to fish them off an abyssal plain, or worse, the bottom of a submarine trench.

In fact, Nature has presented you with an ideal impact area only a few orbits away from your planned impact zone: the Australia's eponymous Nullarbor Plain. It is flat, desolate, and nearly unpopulated - in fact, these very properties already make it one of the world's most popular prospecting grounds for meteorite collectors. Its lack of rainfall and geological stability makes them easy to spot against the background; Mir fragments would be easily spotted as well. You might be thinking that Siberia also possesses these qualities, plus the distinct advantage of being located inside your borders, where the locals are used to falling rocket parts, as opposed to being in another sovereign country. While your local incidence of protest may indeed be lower, alas, in its 51.6 ° orbit, Mir misses most of Siberia. The Nullarbor Plain also beats Siberia because an avid souvenir hunter can prospect in his shirtsleeves and digger hat. His quarry won't get lost in a meter of snow, like your inflatable reentry shield experiment did last year. Aussies have experienced things falling out of the sky before (for instance our Skylab, which didn't hurt anyone) so this won't be such a culture shock. Plus, gambling is a national sport in Australia. Nonetheless, you might offer the Aussies a piece of the action, so to speak, to compensate them for their trouble. This would still leave you hundreds of millions of dollars in the clear. Or, since your industrial ministries are already so skilled at off-the-books bartering, you might be able to arrange a massive beer-for-Mir swap. Beer is their other national pastime.

Now, Mir's estimated reentry ellipse, 6000 km long x 200 km wide, is approximately the same length as the ground track across Australia. In addition to having just 18.5 million people spread across 7.7 million square kilometers ³, Australia is the most highly urbanized country on Earth - 90+% of the population lives in just six big cities on the coastline, on about 0.1% of the land area. Perhaps 80% of the rest are distributed in small towns and cities within 100 km of the coastline. Thus, the interior of the continent is nearly as deserted as an ocean.

You'll probably want to obtain a insurance policy for wrongful death, just in case, but I'm sure one of the Lloyd's Names would cover that bet for a cool million (see below). The premium you have already paid to Lloyd's for $200 million in coverage is grossly conservative and totally wasted; you should have saved the cash and self-insured. Who can possibly be injured under your present plan to rendezvous with the South Pacific - benthic fishes? As for those intrepid boatmen who are choosing to remain in the South Pacific impact area and fish for albacore, I note that although they would have the right to sue for damages under international law, they must first survive the sinking of their boat by your space station and manage to make it to shore, which is only where the international lawyers are to be found. However, they are unlikely to accomplish this feat from the midst of the Roaring Forties or Furious Fifties.

Likewise, for the Australian Option: If a piece of space junk incinerated an Aboriginal nomad in the middle of the Nullarbor Plain, and there were no media around to hear or document it, did anyone really die? Why are we restricting our analysis to taxpayers? Well, koalas, which are already endangered, rarer, and considerably cuter than most people, thus more valuable on a being-for-being basis, nevertheless present smaller targets and are less likely to sue. Kangaroos, on the other hand, are approximately man-sized, warm, and much more numerous, but they do hop around quite a bit and can probably avoid the incoming debris. In any case, since many "roos" end their existence inside dog food cans, we can safely assume that the liability from hitting one is basically nil. (And with the correct collision geometry, they'd still be canned!) Likewise, rabbits in the Outback are all too competent at self-replication, thus their replacement value is also zero. So the signals of these particular nonhuman casualties would be lost in the noise of road-trains. Although some groups such as PETA might decry this attitude toward the dusty denizens of the desert as cavalier, that is a risk you must take.

Recalculating BI for the Australian Option, where:
P_land = 1.00
P_{warm body} = P_hit = 0.00000001, i.e. 1:100 million
P_lethal = 0.5 as before.
So:
P_safe = 0.9995, therefore
P_{kill, j} = 0.0005, or 1:2000

Again, we ignore PD because:

• (a) Mir craters would probably improve the property's value, and
  

• (b) extrapolation based on Australia's 1997 GDP ³ shows that the Outback is worth less than empty ocean (approx. 39 cents/m²).

Cost/Benefit Analysis:
We now have a rational basis for making a decision, based on the expected value (EV) of the alternatives. You can:

• (A) Present plan: Thalassosynchronous orbit. Souvenir value = zero.
  

• (B) Procure an insurance policy for "Bruce Doe" from Lloyd's for $1 million, then deorbit Mir with a Progress flight already paid for ⁴, but direct its fall onto the Nullarbor Plain, then gather up and sell the debris on the international collectors' market. Souvenir value = + US$399 million (minus shipping & handling & commission to the Aussies, plus media and rubbernecking fees).
  

Plan B has much to recommend itself over Plan A, namely a handsome profit margin. Furthermore, in addition to financing a year's worth of your civil space activity with the earnings, Plan B would win you friends among techno-pack rats the world over. So, tovarishch Koptev, do not be dissuaded from this lucrative course by unimaginative killjoys and other officious wowsers! Say Nyet ("Nein") to Plan A from outer space!

1. Current global population estimate courtesy of The Futures Group International, Inc. .

2. various, Statistical Abstract of the United States: 1994 edition, (USGPO: 1994)

3. various, The Time Almanac: 1998 edition, (Information Please Inc., Boston, Mass: 1997), pg. 194.

4. It is a basic principle of engineering economics to ignore sunk costs. See Newman, Donald G., Engineering Economics, 2nd edition, (Engineering Press, San Jose: 1980), pg. 105.

Sep1999; original post to FPSpace listserver
06Mar2001; abbreviated version cited on space.com.
22Mar2001; samizdat on web.

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