Avoiding Nemesis: Does the impact rate for asteroids and comets vary periodically with time?
Is the Earth more likely or less likely to be hit by an asteroid or comet now as compared to, say, 20 million years ago? Several studies have claimed to have found periodic variations, with the probability of giant impacts increasing and decreasing in a regular pattern. Now a new analysis by Coryn Bailer-Jones from the Max Planck Institute for Astronomy (MPIA), published in the Monthly Notes of the Royal Astronomical Society, shows those simple periodic patterns to be statistical artifacts. His results indicate either that the Earth is as likely to suffer a major impact now as it was in the past, or that there has been a slight increase impact rate events over the past 250 million years.
Figure 1: Barringer Crater, also known as Meteor Crater, in Arizona. This crater was formed around 50,000 years ago by the impact of a nickel-iron meteorite. Near the top of the image, the visitors center, complete with tour buses on the parking lot, provides a sense of scale.
Credit: National Map Seamless Viewer/US Geological Service
Giant impacts by comets or asteroids have been linked to several mass extinction events on Earth, most famously to the demise of the dinosaurs 65 million years ago. Nearly 200 identifiable craters on the Earth's surface, some of them hundreds of kilometers in diameter, bear witness to these catastrophic collisions.
Understanding the way impact rates might have varied over time is not just an academic question. It is an important ingredient when scientists estimate the risk Earth currently faces from catastrophic cosmic impacts.
Since the mid-1980s, a number of authors have claimed to have identified periodic variations in the impact rate. Using crater data, notably the age estimates for the different craters, they derive a regular pattern where, every so-and-so-many million years (values vary between 13 and 50 million years), an era with fewer impacts is followed by an era with increased impact activity, and so on.
One proposed mechanism for these variations is the periodic motion of our Solar System relative to the main plane of the Milky Way Galaxy. This could lead to differences in the way that the minute gravitational influence of nearby stars tugs on the objects in the Oort cloud, a giant repository of comets that forms a shell around the outer Solar System, nearly a light-year away from the Sun, leading to episodes in which more comets than usual leave the Oort cloud to make their way into the inner Solar System – and, potentially, towards a collision with the Earth. A more spectacular proposal posits the existence of an as-yet undetected companion star to the Sun, dubbed “Nemesis”. Its highly elongated orbit, the reasoning goes, would periodically bring Nemesis closer to the Oort cloud, again triggering an increase in the number of comets setting course for Earth.
For MPIA's Coryn-Bailer-Jones, these results are evidence not of undiscovered cosmic phenomena, but of subtle pitfalls of traditional (“frequentist”) statistical reasoning. Bailer-Jones: “There is a tendency for people to find patterns in nature that do not exist. Unfortunately, in certain situations traditional statistics plays to that particular weakness.”
That is why, for his analysis, Bailer-Jones chose an alternative way of evaluating probabilities (“Bayesian statistics”), which avoids many of the pitfalls that hamper the traditional analysis of impact crater data. He found that simple periodic variations can be confidently ruled out. Instead, there is a general trend: From about 250 million years ago to the present, the impact rate, as judged by the number of craters of different ages, increases steadily.
There are two possible explanations for this trend. Smaller craters erode more easily, and older craters have had more time to erode away. The trend could simply reflect the fact that larger, younger craters are easier for us to find than smaller, older ones. “If we look only at craters larger than 35 km and younger than 400 million years, which are less affected by erosion and infilling, we find no such trend,” Bailer-Jones explains.
On the other hand, at least part of the increasing impact rate could be real. In fact, there are analyses of impact craters on the Moon, where there are no natural geological processes leading to infilling and erosion of craters, that point towards just such a trend.
Whatever the reason for the trend, simple periodic variations such as those caused by Nemesis are laid to rest by Bailer-Jones' results. “From the crater record there is no evidence for Nemesis. What remains is the intriguing question of whether or not impacts have become ever more frequent over the past 250 million years,” he concludes.
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What are the subtle statistical pitfalls that invalidate previous studies?
In order to test a hypothesis in traditional (“frequentist”) statistics, you posit an alternative called the “null hypothesis”. The null hypothesis should be chosen so as to represent the default situation, for instance: If your hypothesis is that a certain drug can help treat disease X, your null hypothesis would typically be that the drug works no better than giving patients placebos.
Once you have obtained your data, you calculate the probability of obtaining this particular data, assuming that the null hypothesis is true. If the probability is too low (5% and 1% being typical thresholds), you reject your null hypothesis and, in turn, accept your original hypothesis.
However, there are subtle ways of how your choice of original hypothesis can influence your calculation for the probability of your null hypothesis. Notably, when searching for periodicities in impact crater data, you need to take properly into account whether the particular periodicity (say, 13 million years vs. 50 million years) you decide to test against is something you have derived from your data or posited independently of what you have observed. Ignoring this difference can skew your analysis, introducing a bias against the null hypothesis.
Also, in the case of impact craters, there appears to be an underlying trend: an increase in impact rate over the past 250 million years. In the presence of such a trend, the usual null hypothesis assuming constant impact probability is not a valid comparison.
How does the alternative analysis work?
Bayesian inference is an alternative approach to testing a hypothesis that proceeds as follows. To start with, you need different alternative hypotheses. In this case, Bailer-Jones chose constant impact probability; simple periodic (sinusoidal) variations of the impact rate; the case where such periodic variations govern only part of the impact probability; an underlying trend; an underlying trend plus periodic variations.
Before looking at your data, you assume that you cannot decide which of the hypotheses is more probable: you assign an equal prior probability to all of them. Bayesian inference allows you to use your data – in this case the approximate dates or date ranges for different impact craters – to adjust the probabilities. In particular, it tells you how likely each hypothesis is, given the data you have measured. By comparing these probabilities, you can decide which of your initial hypotheses is the most likely, and how much more likely than the runners-up it is.
Bayesian inference is not a cure-all for statistical problems. It has some subtleties of its own, notably concerning the proper choice of prior probabilities. For this particular task, that is, the analysis of time series data with different uncertainties (and, in some cases, upper age limits for a crater only), it is a highly suitable tool that allows well-founded statements about the different hypotheses under discussion. Periodic variation in the cratering rate is strongly disfavoured in all data sets, and there is no evidence for a periodicity added to an underlying general trend. This result is found to be quite robust to the specific assumptions about the priors.
Figure 2: The Nördlinger Ries, or Ries, was formed when a meteor hit the area 15 million years ago. The resulting crater, roughly 20 km in diameter, has since been filled in and eroded. In this natural-colour satellite image, it can just be made out as a circular structure, much less clearly defined than the Barringer Crater, which is significantly younger.
Credit: NASA/J. Allen using data provided courtesy of NASA/GSFC/METI/ERSDAC/JAROS, and U.S./Japan ASTER Science Team