How to Defuse Earth Impact Threat Announcements
Announcements of newly discovered near-Earth asteroids with unusually high Earth-collision chances in the near future have been regularly highlighted in the press and on TV since the 1980s. Recently, they’ve been highlighted on the web. These planetary objects, referred to throughout as NEAs, are formally defined in Figure 1.
Over time, less concern has been noted, unless the announcement concerns a new, relatively big asteroid (hundreds of meters in size) with an alarmingly high probability of hitting the Earth in the next decades.
Impact monitoring and analysis is primarily undertaken by the University of Pisa (the system is called CLOMON21) and NASA-JPL (National Aeronautics and Space Administration Jet Propulsion Laboratory), where it is called SENTRY2. On a daily basis, these two groups collect all astrometric data of all NEAs, new and known, observed the previous night by the observing surveys.
Astrometric data, or astrometric observations, are the optical/radar measurements of the asteroid sky position with respect to the so-called “fixed stars” (the Earth, in the case of radar observations). These data are compiled from all the observatories around the world and made available at the Minor Planet Center (MPC)3 at Harvard University.
Every time additional astrometric observations become available, the characterization of the asteroid orbit improves and the estimated impact probability Vi is re-computed. This may happen in the weeks, months, and even years following the discovery date. A typical pattern is that as the orbit becomes more precisely determined, impact probability Vi often increases initially, but then decreases until it falls to zero, or some very low number.
The reason for the initial increasing behavior is rather technical: Since the uncertainty region generally shrinks with new additional observations, some VI orbital solutions often remain inside the uncertainty region in the elements space.
In the following section, we propose an a posteriori conditional reading of VI impact probabilities. We label our probability as a posteriori since it is obtained as the ratio between two statistical quantities (relative frequency). On the other hand, the VI impact probabilities can be considered a priori, in the sense that they are obtained through sophisticated mathematical models and deductive reasoning (for instance, the choice of an a priori six-dimensional space distribution to sample the uncertainty region).
Statistical Reading of Vi : The Probability W
When a newly discovered NEA is found, a key question is whether the probability that Vi approaches and eventually reaches unity (within this paper, we will use the compact notation ‘Vi → 1’) after the right amount of additional new astrometric observations has become available. This is equivalent to asking whether only knowing that a newly discovered NEA exhibits some VI orbital solutions, what is the probability that Vi will be equal to 1 at the end of the whole orbital refinement process.
The following thought experiment helps motivate this point. Suppose that the existing discovery surveys are able to discover all NEAs that pass close to the Earth down to a size cut-off. This is obviously not true since during their close approaches to the Earth, many unknown asteroids remain too dim to be detected by telescopes. They are too small in size and/or still ‘too distant.’ Moreover, some NEAs are not found because telescopic observations miss them, namely surveys do not image a portion of the night sky when they are there and bright enough to be seen. This becomes especially true for asteroids relatively close to the Sun, where monitoring is more sporadic or even impossible.
We also suppose that every discovered asteroid really impacting the Earth in the future will show some VIs, with low Vi soon after the discovery and fluctuating with an increasing trend as soon as subsequent astrometric observations become available, as usually happens in reality.
In other words, we are putting ourselves in the somewhat idealized situation in which every impacting asteroid above a size cut-off is surely discovered, and monitoring systems surely spot some VIs for it soon after its discovery. Thus, we define W as:
where n(D + ∆D; D) is the number of asteroids with size between D and D + ∆D that actually impact the Earth in the period of time T, with T » 1 year, and v(D + ∆D; D) is the number of asteroids found by monitoring systems among all the newly discovered ones to exhibit VI orbital solutions, in the same size interval and in the same period of time T.
The quantity W can be seen as the a posteriori conditional probability of Vi → 1, and could also be interpreted as a kind of ‘weight’ of the VI impact probability calculation. Now, by dividing both numerator and denominator of (3) by T, we have:
The limit limT → ∞ is the definition of the background annual impact probability, equation (2). The function fvi (D + ∆D; D) = limT → ∞ is the annual frequency of newly discovered NEAs with sizes between D and D + ∆D found with VI orbital solutions.
Note that, according to the earlier assumption, the number n(D + ∆D; D) is counted in the number v(D + ∆D; D), if every impacting asteroid is identified soon after its discovery as having some VI orbital solutions, thus n(D + ∆D; D) is always less than or equal to v(D + ∆D; D).
We focus our attention on (4). Within the hypotheses introduced above, we imagine waiting for a long period (many years), T, and count the number n(D + ∆D; D) of true asteroid impacts on the Earth and the number v(D + ∆D; D) of newly discovered NEAs found by monitoring systems to have VI orbital solutions with size between D and D + ∆D during that period of time. The limit for T → ∞ of the ratio between these two numbers is the conditional probability that a discovered asteroid of size between D and D + ∆D will eventually fall on Earth, given that it has VI orbital solutions. In (4), we have simply rewritten W in terms of the background annual impact frequency ρi and the mean annual VI detection frequency fvi.
Thus, W(D + ∆D; D) gives the probability that an asteroid with VIs (an asteroid inserted into the risk lists of the monitoring systems) has its probability Vi eventually reaching unity.
We noted earlier that W could be interpreted as a kind of ‘weight’ of the VI impact probability calculation. Suppose that, thanks to improvements in observational techniques (e.g., higher positional precision) and orbital computation, the number of newly discovered asteroids identified by monitoring systems as potential impactors decreases in every diameter (or absolute magnitude) range. Accordingly, the probability W will increase (given its definition) in every diameter (or absolute magnitude) range. Since the decrease of the number of potential impactors among the new asteroids means an increased capability in constraining the true potential impactors, the consequent increase of W would equivalently mean an increased capability in constraining the true potential impactors by the monitoring system. Therefore, W could be seen as a ‘weight’ in expressing the actual capabilities of the monitoring system.
Moreover, we can see that W is not directly related to the specific numerical value of Vi, no matter how Vi’s specific, fluctuating numerical figure is. Rather, it depends upon fvi, which, in turn, depends upon observational characteristics. These characteristics are the annual number of NEA discoveries, the amount of astrometric observations available at discovery, the magnitude of astrometric errors (and conventions in their statistical treatment), and the observational geometry and orbital characteristics of the newly discovered asteroids.
Although the value of fvi depends upon contingent, variable features, it is worthwhile to estimate it statistically. Given the total number of NEAs with VIs found at every size between calendar years 2004 and 2009 (Figure 3), an indicative estimate of fvi (D + ∆D; D) is possible (see Figure 4). Consequently, a preliminary estimate of W(D + ∆D; D) can be obtained as in Figure 5. The annual VI detection frequency fvi (D + ∆D; D) shown in Figure 4 has been obtained by dividing the number of all NEAs discovered between calendar years 2004 and 2009 and listed in the SENTRY risk list by six (years), binning the result by size.
Relaxing the optimistic assumptions on the “almost perfect NEA discovery efficiency” and VI monitoring capabilities makes fvi, as approximated with the aid of Figure 4, even a lower limit. As a result, the computation of W(D + ∆D; D) is surely an overestimate.
Furthermore, fvi changes over time, since the discovery completion increases and the discovery rate declines with time (because the NEA population is stable). It is useful to update fvi (and hence W) from time to time. But the sense and the validity of the definition of W are not affected by (and dependent on) such a time dependence.
In summary, the function W(D + ∆D; D) provides us with a first simple tool to evaluate the chance that Vi (the impact probability calculated by the monitoring systems for newly discovered NEAs with VI orbital solutions) eventually reaches unity. As a matter of fact, Vi is a stochastic variable since nobody knows how Vi will evolve with additional observations, and it is perfectly legitimate (and valuable) to define a probability (W) of a (stochastic) probability value (Vi).
Consider now the application of probability measure W to two well-known cases: Apophis and asteroid 2008 TC3 (which actually impacted the Earth the day after its discovery). These examples suggest the reliability of probability W in providing an early direct glimpse of the most likely fate of probability Vi.
Soon after Apophis was discovered in December 2004, impact-monitoring systems identified multiple VIs orbital solutions and obtained the fear-inducing initial impact probability of ~1/38 for the year 2029.
What is the probability W for Apophis, an asteroid with an estimated diameter of nearly 300m? According to Figure 5, W(~300m) is less than 10–6, namely more than four orders of magnitude lower than that initially reported by monitoring systems and close to their current estimate, obtained after some orbit refinement.
Nowadays, it is almost certain that the probability Vi for Apophis will go to zero with future astrometric observations, but it was not so clear at the beginning of the impact-monitoring process (with an initial Vi of ~1/38). As a matter of fact, an early use of probability W would have given a direct glimpse of what would have been the most likely fate of Vi for Apophis.
The same would have happened for 2008 TC3, which was discovered on October 6, 2008, and impacted the Earth about 20 hours later. The impact was predicted by monitoring systems to have a probability of ~100%. Consistently, the probability W for objects in the size range of 2008 TC3 is practically 1, the size of that asteroid being estimated to be between 2m and 4m (see Figure 5).
In the end, the proper impact probability of a newly discovered asteroid is not Vi (which actually fluctuates with new additional observations), but the probability that Vi → 1 (i.e., W).
3www.minorplanetcenter.org/iau/mpc.html. The MPC designates minor bodies in the solar system and has international responsibility for the efficient collection, computation, checking, and dissemination of astrometric observations and orbits for minor planets and comets.
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