I performed a simulation to try to get an answer to compare against a wide range of answers I find on the internet.

Potentially Hazardous asteroids are asteroids that have a minimum orbit intersection distance of 0.05 AU, and an absolute magnitude of 22.0 or brighter, making them at least about 150 meters. But how often do they strike Earth? I hear different figures from different sources. Wikipedia, without referencing its source, claims "once every 10,000 years or less".³ (links at the end of the post.) The Bad Astronomer's blog states "very roughly one such impact every million years or so", referring to impacts causing an explosion comparable to our strongest nuclear weapons.⁴ (Does anyone know what year this blog was from?) A Nasa web site, quoting JPL's NEA Tracking team claims "These bodies impact the Earth only once every 1,000 centuries on average", referring to objects at least 500 meters wide⁵. And a group of scientists studying geological features they believe are caused by ocean impacts put the number at once every 1000 years.⁶ Who's right?

The number of estimated potentially hazardous asteroids (PHAs) is estimated to be as low as 1000 and as many as 2500 ¹. To date, nearly 900 have been found².

But how do you estimate the entire population? Perhaps if we know of nearly 900, and there are an estimated population of 1000, then the discovery rate must be asymptotically approaching 0 by about now, or in a few years from now in the case of the 2500 estimate. Is determining the position of our current count on this curve the type of math model used? Or do we know what percentage should have discovered based on the amount of sky searched over time, given the known sensitivity of our equipment? Or is there another method? I couldn't find any journal papers on the subject. The only info I could find is listed below.

And how do we determine how often we are hit? The BA's blog says "Count ‘em up, apply some math, and you can get a statistical rate of impact". How would you apply the math? It seems like a complicated problem. Each asteroid has a unique orbit. Some are locked into resonance with Earth, which protects them from a collision. Some on the PHA list require a small perturbation to push it onto an Earth-crossing orbit. Some are Jupiter-crossers as well, and could be ejected from the solar system. Many PHAs have steeply-inclined orbits.

I tried an n-body simulation and concluded that we're hit about once every 50 thousand years on average if the population is 2500, or once every 130 thousand years on average with a population of 1000. Feel free to critique my method:

Using Gravity Simulator, I modeled the solar system, including the 8 planets and the 794 known PHAs as of July 2006. The position and velocity vectors of the planets and the PHAs were supplied by Jet Propulsion Laboratory's Horizon Ephemeris service. A quick test run of the simulation revealed that simulating a system with this many objects, at a time step slow enough to allow collisions would take hundreds of years of computer time to simulate 1 million years, the high-end impact rate estimate. So rather than directly modeling this scenerio, I increased Earth's radius 100 fold. A larger Earth will get hit more often, and it allows me to take a higher time step since objects can not simply step over such a large Earth. By increasing the Earth's radius, I am increasing the area of Earth's cross section that it will present to an incoming asteroid by the square of the increased diameter. Thus, by increasing Earth's radius 100 fold, the Earth should get hit 10,000 times as often.

My simulation reveals that a PHA should impact Earth once every 50,000 years on average for a PHA population of 2500, or once every 130,000 years on average for a population of 1000 PHAs. Here's how I arrived at the number: After 1341 years, largeEarth suffered 78 impacts. But the simulated population is less than the estimated population. Since the simulated population decreases by 1 each time there is a collision, the simulated population is expressed as Initial simulated population - 0.5 * impacts, yielding an average simulated population over the course of the simulation. Dividing an estimated population of 2500 PHAs by an initial simulated population of 794 yields 2500 / (794-0.5*78) =3.311. Multiplying this by the 78 recorded impacts reveals that Earth should have been struck 258 times. Dividing the number of simulated years by this figure yields 1341 / 258 = 5.1977. Earth should get hit once every 5.1977 years. Finally,multiplying this number by Earth's exaggerated size of 100 Earth Radii squared gives us an impact frequency of 52,000 years.



I: average impact frequency (in the same units of time used for T)
R: radius (real Earth and exaggerated size Earth)
T: time (initial and final)
N: number of asteroids (estimated and simulated)
i: number of simulated impacts recorded

(R_bigEarth / R_Earth)² * (T_f-T_i) / ((N_est/(N_sim-0.5*impacts))*impacts)

Plugging in numbers for an estimated population of 2500 PHAs, 794 simulated PHAs, 78 impacts, a start year of 2007, and an end year of 3348 gives:
100^2*(3348-2007)/((2500/(794-0.5*78))*78) = 51920.7692307692, or an impact frequency of 52 thousand years

Using an estimated population of 1000 PHA
100^2*(3348-2007)/((1000/(794-0.5*78))*78) = 129801.923076923, an impact frequency of 130,000 years

I suspect that the impact frequency is slightly lower by a few percent, as Earth's gravity will pull in a few that were otherwise destined to miss my a few hundred kilometers. The exaggerated Earth with its weaker surface gravity is less efficient at doing this.

Screenshot. The solar system including nearly 800 asteroids classified as Potentially Hazardous:



[1] http://www.francisquito.org/why_nea.htm , and other sources
[2] http://neo.jpl.nasa.gov/neo/pha.html , and other sources
[3] http://en.wikipedia.org/wiki/Potenti...rdous_asteroid
[4] http://www.badastronomy.com/bablog/2...y-space-rocks/
[5] http://imagine.gsfc.nasa.gov/docs/as...rs/danger.html
[6] http://library.lanl.gov/tsunami/213/scheff.pdf
[7] http://impact.arc.nasa.gov/downloads...dmfh101200.pdf

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www.gravitysimulator.com

It's very difficult. There are several search programs which look for PHAs. Each search program has its own idiosyncrasies -- this one never looks within 90 degrees of the Sun, that one goes to 19'th magnitude for fast-moving objects but 21'st magnitude for slow-moving objects, etc.. One must figure out the selection effects in each search program in order to convert the number of objects they actually observe to the number of objects which actually exist. That's very tough.

One problem, for example, is that big, bright objects are easier to find ... but small, faint objects are intrinsically more common (usually). So, you might find 10 big PHAs and 20 small ones. Does that mean that the true population is roughly 1/3 "large" and 2/3 "small" bodies? Or does it mean that the true population is more like 1/10 "large" and 9/10 "small", but you preferentially discovered more of the big, luminous ones? Probably the latter.

Astronomers try to make reasonable models of the size distribution of the real population: for example, a power-law with slope -2. Then, they can compute the total number of objects in the population based on the number they actually observe. However, if they make even a small error in the model of the population -- the power law has a slope of -2.5 -- then the total number of objects can be very different from their estimate.

As for your simulations of impacts on the Earth ... it looks reasonable. Would you like to check your results? You can, you know. Calculate the impact rate on Mars. Then start looking at the surface of Mars (or read papers on the cratering rate), and see if the number of fresh, young, big craters within some estimated time interval agrees with your simulation. Mars will be a better target than Earth, since it has no oceans and relatively little erosion. We should be able to find most of the big recent impacts, given the global coverage of the surface from several space missions.

Nice work!

I notice that the current risks site:

http://neo.jpl.nasa.gov/risk/

now has no objects seen in the last 60 days; it used to have quite a few.

Does this mean that there are not so many new ones to find now?

Following SM's suggestion, do you have the Moon in Gravity Simulator too tony873004?

If so, what's the relative PHA impact rate (Earth vs Moon), from your simulations?

Independently, does the reconstructed cratering history of the Moon provide any useful upper or lower (or both) bounds?

Of course, an inconsistency somewhere may be a signal of something else - a pulse of new PHAs perhaps (an asteroid collision?), or a previously undiscovered stream. As SM indicated, estimates of the population are based on a set of assumptions ... one such is that distributions tend to be smooth ... if the relevant part of the universe is a bit non-conformist, then it's back to the drawing board.

In this particular simulation there is no moon, as I exaggerated Earth's size by 100x. The moon got swallowed But in most Gravity Simulator simulations, the Moon is included.

But it is easy to compute how many times the Moon would have been struck. It is about 1/4 the diameter of the Earth, so it presents a cross-section of (1/4)^2 or 1/16 the size of Earth's. So my model would suggest that the Moon should be struck (50,000 * 16) = once every 800,000 years for 2500 PHAs, and 130,000*16 = once every 2 million years for 1000 PHAs.

The problem I see with this method is I don't know the dates of the craters on the Moon or Mars. The majority of them probably were formed during a period of much heavier bombardment. I wouldn't know how to get a count of all craters newer than say 100 million years.

In a sense, comparing my figure to the figure obtained by the geologists counting ocean-floor craters that they claim caused tsunamis which left geological imprints on land, is a way of double checking. But the double check fails. Their answer of once every 1000 years and my answer of once every 50-130 thousand years is more than an order of magnitude different.

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And now the current risks site:

http://neo.jpl.nasa.gov/risk/

only has, many, objects seen in the last 60 days. I do now wonder to what extent this site is a reliable report of the actual database of objects.

There are some geologists who do try to figure out the (rough) dates of craters on the Moon and Mars. The interested reader can start with these references

http://www.sciencemag.org/cgi/conten.../314/5805/1573
(for _really_ recent craters, but too few for our purposes here ....)

http://www.psi.edu/projects/mgs/cratering.html
(a more relevant introduction)

and, if sufficiently interested, use the ADS abstract service
to find papers in the technical literature. Unfortunately, many are likely to be published in the journal "Icarus", which does not permit free access to much material at all, even older stuff. Sigh.

I'd just study the Moon and account for the effects of Earth gravity.

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I am Mugs, of the Alien clan of Usa, Nordamerica, a Terran, of Sol.

Mine: "Perception isn't reality. It's merely an abstraction thereof, and quite often not a very good one at that."

Heinlein's: "Staying young requires the unceasing cultivation of the ability to unlearn old falsehoods." "Freedom begins when you tell Ms. Grundy to go fly a kite."