Predicting Inter-Asteroid Collisions

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

An astronaut standing on the far-side of the moon would feel lighter, not heavier, because of the earth"s attraction on the other side, for the same reason that there are two high tides per rotation, not one. However, it should be notated that tides are static, not cyclic: we only think that they are cyclic because our planet is rotating underneath them; on the moon, locked into keeping the same face towards the earth as it orbits, its high tides are permanently at the centre of the near-side face and at the centre of the far-side face. Meanwhile, for an astronaut standing on the far-side of the moon, the experience is a tame version of what it feels like to undergo spagettification: nowhere near as dramatic as if he were floating close to the event horizon of a black hole, but still in the foothills of the same effect. High tide is precisely the effect of the gravity field starting to pull particles away from the parent body. The molecules on the side facing towards the other body find they are orbiting the Bary Centre too slowly, and are trying to drop towards it, while the 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. On the earth, it is the water molecules being pulled away; on the moon, the astronaut is being tugged away from being part of the moon-astronaut system, and experiences the effect as the two bodies becoming less attached (lighter).

Algerbraic astrolabe

Aveni (2008) describes how the Pointer stars of Ursa Major can be read as the hour-hand of a clock, with Polaris as the centre of the clock face. There are three complications, though:

• It is a 24-hour clock face
• The hour hand progresses anti-clockwise
• Midnight is not necessaruly at the top of the clock face.

The last of these requires the clock face to be calibrated. It turns out that a good starting point is to note that the hand is pointing directly up at midnight at Greenwich on 10th March. Adjustments can then be made by subtracting an hour from this time for each 15 degrees that the location has a longitude to the east of Greenwhich, adding on the number of hours that the location is ahead of Greenwich Mean Time, G, and subtracting two hours for the number of months, M, since the reference date. It turns out, then, that the hand is pointing directly up at midnight at La Tour d'Aigues (L=5°33' and G=1) on 20th March. There remains one further offset to take into account: the year does not start on 10th March, nor on the first day of the first month, but on 30th Nuvember (the zeroth day of the zeroth month). This offset turns out to be 6.7, or 30.7 once a further 24 hours have been added in to make sure the formula comes out with positive values.

Treating the pointer stars like the hand of a clock, a reading of 10 o'clock is two hours before midnight on a normal clock face, so four hours after the calibrated time on the star-based cycle; and, correspondingly, a reading of 2 o'clock is two hours after midnight on a normal clock face, so four hours before the calibrated time on the star-based cycle.

Putting this all together as a pseudo formula for Excel:

T =MOD(P-2*M-(L-G)+30,7;24)

where P is the position in the sky, on the notional 12-hour clock face, of the pointer stars (for example, the half-past four position would have P=4.5), M is the month (where each day counts as a thirtieth of a month, and 10th March would be 3.33333), L is the longitude east, divided by 15, and G is the number of hours ahead of GMT for the current location. This L-G component is the reason why this method cannot be used as a means of determining the longitude of the current position. The inverse function, to predict the position of the pointer stars is then:

P =12-MOD(T+2*M+(L-G)-30,7;24)/2