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Predicting Inter-Asteroid Collisions

The Minor Planet Center (MPC) of Harvard University maintains a public, on-line database of the orbit parameters of all the known asteroids and comets (well over 65000 of them).

It is inevitable that collisions occasionally occur between these objects. Indeed, there is evidence of this having occurred, by the existence of craters on some of them, so-called "rubble-piles" as the constitution of others (253, Mathilde, for example), and the existence of at least one (243,Ida) that has another as a satellite (Dactyl) with a completely different composition. Further evidence is provided by the fate of Shoemaker-Levy-9, and the more recent craters of the Moon, Mercury, etc, that such events do still occur (albeit with greater probabilities for the larger bodies).

The asteroid belt has existed for at least most of the 4.6 billion year age of the solar system. Thus, we might assume that, overall, the asteroid belt is in a state of dynamic equilibrium. That is, collisions are causing objects to combine or to break-up at roughly equal rates. (The alternative is that the composition of the asteroid belt has evolved over that time, but with an extremely long time constant.)

A collision between two objects is guaranteed to be an interesting event, and an enormous opportunity to collect observational data that would not normally be available. For example, it is possible to observe the material that is thrown up by the collision, the duration of the aftermath of the event, the sizes of the resulting objects (unified or fragmented from the parent bodies). Even a near-collision still yields a wealth of information, via the perturbations to their respective orbits, giving the masses of the two objects involved, and perhaps some insight on their internal mass distribution.

The aim of this project, therefore, is to use the information in the MPC database to predict, in advance, when and where the next collision is likely to occur, so that preparations can be made sufficiently in advance for the observation of the event.

Extent of the Prediction

Given that collisions are likely to be rare, and to involve two objects of small size and obscure nature, it is likely that the next collision will not be detectable by optical instruments on earth. It follows, therefore, that the prediction must be sufficiently long term for it to be possible to prepare a space mission to travel to observe it. At the very least, therefore, the predictions must be made about 20 years in advance (10 years to plan and build the craft, and 10 to travel to the site).

Moreover, the prediction must be made at a very high accuracy, and with a very high degree of confidence, if the potential funding bodies are to be convinced that the collision really is going to occur when the spacecraft arrives at the site. For this, we need to consider two sources of error: the reliability of the initial data, and the reliability after the repeated calculations in the simulation.

For the latter, we can immediately note that the 20-year simulation period represents only three orbits of an outer asteroid like (87,Sylvia), and only six orbits of an inner asteroid like (951,Gaspara).

Theoretical Feasibility of the Project

There are more than 216 objects in the MPC database.

Within each simulation cycle, there are three internal steps:

  1. There are 216 two-body orbit calculations to perform, using Kepler and Newtonian laws, applied to simple 2D ellipse-based geometry suspended in 3D space.
  2. Corrections must be applied for each body that violates the two-body assumption (because two of them have approached each other too closely, or because of the influence of the major planets).
  3. The asteroids are taken in pairs, to see if the distance between them is interestingly small. Each comparison is, in fact, very simple (about 10 floating-point operations), but there are over 232 of these to perform. Indeed, in terms of computation time, step-3 dominates step-2 and step-1 in the ratio of about 6667:100:1.

Since the MPC database is updated daily (albeit only for selected entries each time), we might aim for a simulation time-step of 1 day. Thus, there are 7305 time-steps in the 20-year prediction. Given that it is, in principle, possible to buy a supercomputer that is capable of 100 Teraflops, the project appears at least to be possible; and in fact even with a 10 Gigaflops computer, the full simulation would take about 9 hours.

Practical Feasibility of the Project

Most computing problems can be characterized either as being compute-intensive, memory-intensive or IO-intensive, or some combination. The present one falls into the category of purely compute-intensive. Use of a cheaper computer merely scales the simulation time linearly with the compute speed. Thus, a 10 Megaflops computer will take about 9000 hours (about 1 year) per 20-year simulation.

Even this is still feasible (though it would require some careful design, either of the uninterruptible power supply, or of the resumable software). Moreover, by restarting the simulation immediately after the previous run has completed, we would get an annual update of the predicted collisions within the forthcoming 20-year period. However, there are many tricks that we can use to improve on this already acceptable situation.

Even if every asteroid were as massive as (1,Ceres), it would take of the order of four days for an asteroid to cover the distance between being too far apart, and being close enough to exert a gravitational pull one millionth as strong as that experienced towards the sun; that is, for the two-body assumption to be out by one part per million (1ppm). Thus, the time step need not be daily, but once every four days, thereby allowing the compute-time to be reduced to 3 months.

But there are further tricks that can be used to reduce the simulation time. For instance, since the compute-time is dominated by the process of comparing the positions of pairs of asteroids, we could consider representing the finite volume of space directly as voxels within the computer memory. At each time step in the simulation, we could clear all the voxels, and then run the calculations to find out where each asteroid will be next, and marking the corresponding voxel as occupied. If an attempt is made to mark any given 3x3x3 block of voxels twice, it means that two objects are close enough to be interesting, and to warrant closer analysis.

Since most of the asteroids orbit within a torus 2.85 AU in radius, and 0.64 AU in cross-sectional radius, we could adopt polar coordinates, and represent the voxels as blocks that are 0.046 AU by 0.046 AU by 1.2 degrees of arc. (1.2 degrees of arc is 0.046 AU at the orbit of Gaspara, and 0.073 AU at the orbit of Sylvia.)

Under this scheme, the torus (which has a volume of 23 AU3, or 8x1034m3) has a volume of 180000 voxels (within a rectangular band that is 1.28 AU by 1.28 AU by 360 degrees, containing 1,440,000 voxels). Using this hashing method, the simulation cycle time can be greatly reduced, so that the whole 20-year simulation can be completed within 2 days, albeit at the expense of increasing the amount of memory needed by the computer.

Alternatively, we could conduct all of the orbit computations in polar coordinates, rather than the Cartesian ones assumed so far. Step-2 and step-3 need only involve a single floating-point comparison at first (on the angular coordinates of the two objects) to see if a more detailed calculation is warranted. Since this divides the time of step-3 by almost 10, it gives a simple way of dividing the 92-day simulation time down to 10 days.

Allowing for Near-collisions as being Interesting Events

Given that there is no lower limit on the definition of the size of an asteroid, we can imagine that grains of sand are colliding with each other and with bigger bodies quite often. However, we have set our own arbitrary definition on the size of an interesting event as being a collision, or near collision, between two bodies that are listed in the top 65536 asteroids.

One question that arises is that of what qualifies as a near-collision.

It was stated earlier that the value of near-collisions is that they allow the perturbations in orbits of the two bodies to be measured.

If modern instrumentation is up to the task of detecting a 1% perturbation, we could define this as the minimum condition for the space mission to be exciting enough for us to be able to sell it to the funding bodies.

This, of course, is the same sort of calculation as has already been contemplated. The violations of the two-body assumption were assumed to have occurred when the solar orbit was perturbed by 1ppm. Now, we are interested in looking for perturbations of 1%, or 10000ppm. At 1ppm, the distance from an asteroid of the mass of Ceres was 0.046 AU. At 10000ppm, it would have to be as close as 69000 km.

We could consider representing the same finite volume of space as mini-voxels, 100x100x100 times smaller than the ones before, that are each 69000km by 69000km by 43 seconds of arc. (43 arc seconds is 69000km at the orbit of Gaspara, and 110000km at the orbit of Sylvia). Under this scheme, the torus has a volume of 1.8x1011 mini-voxels. (As we have already seen, this is only one of many possible simplifications that could be used. In practice, we could use the same polar coordinate hashing as was used before.)

Probable Pay-off of the Project

Earlier, it was stated that 20 years is the minimum period in which we would hope to be able to predict the next collision. We would like the maximum period, too, to be of this order. That is, even if we have just missed one collision, the next one will be conveniently soon afterwards.

The voxel approach also gives a method for estimating the number of near-collisions that might be expected within the 20-year simulation period. The probability of finding any given 3x3x3 block of the mini-voxels, centred on any of the 216 asteroids chosen at random, to contain at least one more asteroid is 3.6x10-7. The probability of finding an interesting event in the entire system approaches 0.632 (1-1/e) after just over 100000 repeats of the experiment, which is about 1157 years when the 4-day time-step is assumed between experiments.

In fact, the above statistical analysis needs a bit more work on it, but this will have to be left as "plans for future work". It does suggest, though, that the instrumentation would have to be capable of detecting a 0.1% perturbation in the orbits, during a near collision, for the probability of finding such an event to approach 0.632 during only 37 years.

Furthermore, not all of these hits will turn out to interesting. To have two asteroids within a 3x3x3 cube of mini-voxels (one in the central mini-voxel) could mean that they are 2√3 times further apart than assumed so far. And, assuming that neither of them is Ceres, the gravitational attraction between them will be much less than the 0.1% of that towards the sun that was set as the target.

Moreover, the earlier analysis assumes a uniform distribution of the asteroids within the torus, and a random movement of the asteroids between the experiments, which is manifestly not the case. It does, though, at least set a ballpark figure on the probability of finding an interesting near-collision. At the very least, the asteroids will have stirred around by a few hundred mini-voxels between two successive experiments. Even so, a better analysis must eventually be attempted.

However, despite all these caveats, the simulator would at least be capable of picking out the potentially interesting events. As soon as one has been found, it can be simulated in more detail, to investigate whether it really is as interesting as it at first appears.

Confidence in the Predictions

Eventually, I will need to say something more about the accuracy of the starting data (many entries only having their parameters updated occasionally from observation), and the accuracy after a 1822-step simulation. Will the spacecraft find each of the candidate asteroids, where it expects to find them, after its 10-year voyage? The risks of false-positives and false-negatives.

In any case, as has been stated earlier, the main aim of the project is to identify potentially interesting near-collisions. Once identified, a more detailed study can be focussed on that one event, to double check the accuracy of the prediction, and to judge the closeness of the approach of the two bodies.

Conclusion

It is concluded that the project is feasible, it is also necessary to ask whether another research group has already undertaken the project.

Since the project relies so heavily on the data in the MPC database, the research group at Harvard University would be one of the main places to ask before embarking on the project. In any case, their permission ought to be sought before making such extensive use of their database.

Campaign against Comet Earth

It ought to be a concern that though thought to have established early after the formation of the Earth (NS, 08-Aug-2015, p17) we still do not know when the magnetic field was first established (NS, 14-Jan-2017, p33), how long the magnetic field is absent when it occasionally flips in polarity, and that we do not yet understand how the geodynamo establishes itself in the turbulence of the molten terrestrial core (NS, 07-Jan-2017, p10; NS, 25-Aug-2001, p24; NS, 10-Feb-2007, p14); and similarly those of, or lack thereof, of Mercury (NS, 12-May-2007, p20; NS, 16-May-2015, p19) or Mars (NS, 07-Jan-2017, p12; NS, 10-Feb-2001, p4).

At one astronomical unit distance from the sun, the earth would make a wonderful comet, venting its atmosphere into a glorious tail, that would last of the order of a couple of thousand orbits. What prevents this from happening is the earth's magnetic field, deflecting the solar wind. This field is generated by the internal dynamo of the earth's molten core. It is true that the proportion of energy that geothermal is taking out of the core is insignificant (in comparison to the amount of energy stored), and insignificant, too, in comparison to the amount already dissipated to seismic and volcanic activity. Thus, present uses of these energy stores are insignificant; but, we know that human greed will inevitably change this if we allow such an energy source to continue to be developed. This is in the same way that mankind, before the industrial revolution, was similarly making hardly any impact on global carbon dioxide levels, and yet was laying the foundations for today's society getting those levels up from 0.03% to 0.04% within my lifetime. It is true that the thin end of the wedge would be extremely long if we only needed to worry about domestic use, but military and industrial uses would have more scope. However, it is true that that heat, from the crust, would have been destined for transfer to the atmosphere, anyway.

Tidal forces

The usual story is of why, on the earth, there is a high tide on the side facing away from the moon, as well as the side facing towards it. The water molecules on the side facing towards the moon find they are orbiting the Bary Centre of the Earth-Moon system too slowly, and are trying to drop towards it, while the water molecules on the other side find they are orbiting the Bary Centre too quickly, and are trying to raise themselves up into a higher orbit. In effect, those water molecules are feeling lighter than those in the rest of the oceans.

Similarly, if you stand on the far side of the moon, you would feel lighter because of the position of the Earth. The molecules in your body (and hence your whole body) would be trying to raise themselves up into a higher orbit. The tidal forces would be effectively be trying to break apart the Moon-Human system, though not succeeding, in the case of standing on the surface of the Moon, so you only feel lighter, not actually drifting off into orbit.

This makes the explanation of tidal forced more intimate, by framing the story in the 2nd person rather than the usual 3rd person description in the passive voice. In addition, it uses an example that illustrates that tidal forces have nothing to do with the rotation of the body (we are used to two high tides a day on the Earth, but for the Moon the tidal forces are stationary, but stil potentially disruptive).

Algerbraic astrolabe

Based on the description on p74 of Aveni (2008), the equation for telling the time from the night sky, in the northern hemisphere, can be generalised to: 2p+2m+t+e=32, where: t is the current time, on the 24-hour clock; e is the number of hours that the current time, in the current time zone (including daylight saving time, if appropriate), is ahead of local time (negative if it is behind); m is the month of the yeqr; and p is the position of the pointer stars, in Ursa Major, read from an imaginery clock face, with the pole star as the centre of the clock. For this, it is convenient to handle fractional values in base-60, counting the date of the month, for example, as 2 sixtieths of a month.

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