# Extinction Level Event (Asteroid Impact Hypothesis) Likelihood Equation

We have the Drake Equation that calculates the probability of intelligent ET (extraterrestrials/aliens).

Since the asteroid impact hypothesis has been more or less accepted as the cause of the K-Pg Extinction Event that wiped out the nonavian dinosaurs, I was wondering if there's a Extinction by Asteroid/Comet Likelihood Equation along similar lines as the Drake Equation.

A crude formula might look like this:

$$N$$ = Number of asteroids/comets
$$f_I$$= Fraction of N that regularly visit the inner solar system
$$f_P$$ = Fraction of $$f_I$$ that are planetkillers
$$f_k$$ = Fraction of $$f_P$$ that has a killzone that includes the earth
Probability of extinction by asteroid/comet = $$P(E_{a/c}) = f_I \cdot f_P \cdot f_K$$

Number of threats = $$f_I \cdot f_P \cdot f_K \cdot N$$

Knowing that there are X asteroids that could threaten is not in itself useful. What you want to get is a event rate, the probability of something happening per year.

One oversimplified way is to reason like this: each dangerous object sweeps out a "risk volume" $$V=\pi b^2 v$$ per unit of time, where $$b\sim R_\oplus$$ is the distance between the object and the Earth that would lead to an impact, and $$v$$ is the velocity. The probability per unit time that Earth is in any volume if there are $$N$$ objects is $$P = 1-\exp(-\frac{\sum_{i=1}^N V_i}{V})\approx 1-\exp(-N\bar{V}/V)$$ where $$V_{solar}$$ is the relevant volume of the solar system (about $$10^{26}$$ m$$^3$$ if we use 30 AU). So you could now use $$\bar{V}\approx 10^{18}$$ or so (depending on your views on $$b$$ and $$v$$) and start estimating $$N$$ using your formula.

This is somewhat doable, but quickly get complex (different populations, orbits are not actually evenly distributed, etc.) A better approach may simply be to look at the past impacts causing extinctions! Depending on how you count, there has been one known mass extinction due to impacts since the Cambrian 538.8 mya, so the rate might be on the order of $$2\cdot 10^{-9}$$ per year. But that likely leaves out a fair number of minor extinctions. If we assume all Big 5 were due to impacts the rate becomes $$9\cdot 10^{-9}$$ per year.

Incidentally, to get these values in the above formula for the assumed values, $$N$$ should be in the range 0.2 to 1.

Obviously this can be improved: we can use statistical modelling to get error bars, we can use the known size distribution of asteroids (a power law) to estimate the fraction of Earth-crossers and long-periodic comets that could be bad and their inflow rate, and so on. But that misses the Drake equation approach of trying to find a quick-and-dirty model that shows the key variables we care about and might want to estimate.

• Asante sana (thanks a lot). Here's what appears doable in principle with the aid of a supercomputer. Take the Chixculub crater in Mexico, big enough to have been the dinosaurs killer. It's in a particular spot, but that's not quite enough. We need to know, how shall I put it?, its astronomical/celestial position (factoring in earth's axial rotation and circumsolar revolution) when the asteroid struck. From crater features, we can estimate the angle of attack. From these data points, can we not calculate the period of the associated asteroid family ... contd. Commented Nov 13, 2023 at 17:13
• I (only) feel it has to be cluster of planet-annihilators in orbit around the sun and regularly crossing earth's orbit. In other words, the family may be returning 🤔 Commented Nov 13, 2023 at 17:15
• @AgentSmith - Huh?None of this has anything to do with the question and answer. Commented Nov 13, 2023 at 17:19
• @AgentSmith - The first question: there are papers on this, but the uncertainties are necessarily huge. The planet annihilator idea has no support. 65 million year orbits are possible in theory, but so remote that passing stars will disturb them. Commented Nov 13, 2023 at 18:05
• @AgentSmith - It feels like you want to ask other questions rather than do this as comments. And yes, a huge asteroid can still do a lot of damage at a slow speed. But the energy grows linearly with mass and quadratically with speed. Commented Nov 14, 2023 at 14:19