<p>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.</p>
<p>One oversimplified way is to reason like this: each dangerous object sweeps out a &quot;risk volume&quot; <span class="math-container">$V=\pi b^2 v$</span> per unit of time, where <span class="math-container">$b\sim R_\oplus$</span> is the distance between the object and the Earth that would lead to an impact, and <span class="math-container">$v$</span> is the velocity. The probability per unit time that Earth is in any volume if there are <span class="math-container">$N$</span> objects is <span class="math-container">$P = 1-\exp(-\frac{\sum_{i=1}^N V_i}{V})\approx 1-\exp(-N\bar{V}/V)$</span> where <span class="math-container">$V_{solar}$</span> is the relevant volume of the solar system (about <span class="math-container">$10^{26}$</span> m<span class="math-container">$^3$</span> if we use 30 AU). So you could now use <span class="math-container">$\bar{V}\approx 10^{18}$</span> or so (depending on your views on <span class="math-container">$b$</span> and <span class="math-container">$v$</span>) and start estimating <span class="math-container">$N$</span> using your formula.</p>
<p>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 <span class="math-container">$2\cdot 10^{-9}$</span> 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 <span class="math-container">$9\cdot 10^{-9}$</span> per year.</p>
<p>Incidentally, to get these values in the above formula for the assumed values, <span class="math-container">$N$</span> should be in the range 0.2 to 1.</p>
<p>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.</p>

CC BY-SA 4.0
2023-11-13T17:13:04.790Z
*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.
Hudjefa
CC BY-SA 4.0
2023-11-13T17:15:04.823Z
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* 
Hudjefa
CC BY-SA 4.0
2023-11-13T17:16:50.593Z
Can an asteroid have a period of $65$ million years? Do celestial *forces* (gravity/etc.) permit/allow such, what has to be, gigantic orbits?
Hudjefa
CC BY-SA 4.0
2023-11-13T17:19:20.110Z
@AgentSmith - Huh?None of this has anything to do with the question and answer.
Anders Sandberg
CC BY-SA 4.0
2023-11-13T17:26:50.567Z
Just trying to see how far science can take us. Can you answer the questions?
Hudjefa
CC BY-SA 4.0
2023-11-13T18:05:41.847Z
@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.
Anders Sandberg
CC BY-SA 4.0
2023-11-14T02:06:45.810Z
I forgot to consider *speed*, but asteroids that are spoken of (in the media) are usually reported to travel at extremely high speeds. If an asteroid were gigantic but *unusually/relatively* slow, it could still do damage ($\frac{1}{2}mv^2$). Right?
Hudjefa
CC BY-SA 4.0
2023-11-14T14:19:45.890Z
@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.
Anders Sandberg
CC BY-SA 4.0
2023-11-14T15:42:12.673Z
*Asante sana*. Most interesting. That's odd, don't you think, considering, rather *nebulously*, $F = ma$ (linear for both mass & acceleration). Does *shape* matter as well? For instance, will a "pointy" asteroid do more damage than a roundish one? I guess I'm talking about *pressure* instead of force here.
Hudjefa
CC BY-SA 4.0
2023-11-13T16:59:41.820Z
2023-11-13T16:59:41.820Z
Anders Sandberg
<p>We have the <strong>Drake Equation</strong> that calculates the probability of intelligent ET (extraterrestrials/aliens).</p>
<p>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 <em>Extinction by Asteroid/Comet Likelihood Equation</em> along similar lines as the Drake Equation.</p>
<p>A crude formula might look like this:</p>
<p><span class="math-container">$N$</span> = Number of asteroids/comets<br />
<span class="math-container">$f_I $</span>= Fraction of N that regularly visit the inner solar system<br />
<span class="math-container">$f_P$</span> = Fraction of <span class="math-container">$f_I$</span> that are planetkillers<br />
<span class="math-container">$f_k$</span> = Fraction of <span class="math-container">$f_P$</span> that has a killzone that includes the earth<br />
Probability of extinction by asteroid/comet = <span class="math-container">$P(E_{a/c}) = f_I \cdot f_P \cdot f_K$</span></p>
<p>Number of threats = <span class="math-container">$f_I \cdot f_P \cdot f_K \cdot N$</span></p>

<asteroids><comets><extinction>
Extinction Level Event (Asteroid Impact Hypothesis) Likelihood Equation
Hudjefa
# Extinction Level Event (Asteroid Impact Hypothesis) Likelihood Equation

(Asked by: username_0 on 2023-11-13)

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 = $f\_I \cdot f\_P \cdot f\_K \cdot N$

**Tags:** asteroids, comets, extinction

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## 1 answers

### Answer by username_1 on 2023-11-13. Score: 5 | Accepted answer

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.

> **Comments:**
> **username_0**: *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.  
> **username_0**: 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*   
> **username_0**: Can an asteroid have a period of $65$ million years? Do celestial *forces* (gravity/etc.) permit/allow such, what has to be, gigantic orbits?  
> **username_1**: @AgentSmith - Huh?None of this has anything to do with the question and answer.  
> **username_0**: Just trying to see how far science can take us. Can you answer the questions?  
> **username_1**: @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.  
> **username_0**: I forgot to consider *speed*, but asteroids that are spoken of (in the media) are usually reported to travel at extremely high speeds. If an asteroid were gigantic but *unusually/relatively* slow, it could still do damage ($\frac{1}{2}mv^2$). Right?  
> **username_1**: @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.  
> **username_0**: *Asante sana*. Most interesting. That's odd, don't you think, considering, rather *nebulously*, $F = ma$ (linear for both mass & acceleration). Does *shape* matter as well? For instance, will a "pointy" asteroid do more damage than a roundish one? I guess I'm talking about *pressure* instead of force here.  

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