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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.
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.
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).
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.