Earth Versus Catastrophic Meteor

Question

How evasive is the Earth to Catastrophic Meteors?

Google says the Earth is approximately 92.96 million miles from the Sun. It also says the suns radius is 432,474 miles.

Therefore, it is 93,392,474 miles from the Earth to the center of the Sun:

92,960,000 + 432,474 = 93,392,474 miles

I've heard the Earth's orbit is elliptical, but to keep things simple, let's just say the orbit is a perfect circle. Therefore the radius of this circle is 93,392,474 miles. This would mean the diameter of our orbit would be 186,784,948 miles:

93,392,474 x 2 = 186,784,948 miles

The circumference of a circle is π x the diameter of a circle. Therefore, the number of miles in Earth's orbit would be about 586,504,736.72 miles:

3.14 x 186,784,948 = 586,504,736.72 miles

So, the Earth travels about 586,504,736.72 miles in one year. Since there are 365 days in year, that would mean that Earth travels about 1,606,862 miles in one day.

586,504,736.72 ÷ 365 = 1,606,862.292383562

That would mean the Earth travels about 66,953 miles per hour:

1,606,862.292383562 ÷ 24 = 66952.595515982

And this would mean the Earth travels about 1,116 miles per minute:

66952.595515982 ÷ 60 = 1115.876591933

Lastly, this would mean the Earth travel about 18.6 miles per second:

1115.876591933 ÷ 60 = 18.597943199

Google says the diameter of the Earth is 7,917.5 miles. So, as fast as the Earth is traveling, a catastrophic meteor has about a 7 minute window to hit target Earth.

7917.5 ÷ 18.597943199 = 425.719119329 seconds
425.719119329 ÷ 60 = 7.095318655 minutes

What are the odds of a catastrophic meteor hitting the Earth? Or, maybe a better question: What are the odds of the Catastrophic Earth hitting a poor meteor!?

Answer 1

There is nothing wrong with your maths, but it isn't relevant to the question of the frequency of impact.

Now you ask about comets, but do you also want to include asteroids, since these are rather more common.

You also ask about catastrophic, and this is a poorly defined word. A Tunguska scale event is a local disaster, but has no significant global effects.

The Torino scale classifies objects according to their threat. A "10" on the Torino scale signifies a object carrying an energy of more than 10000 Megatons of TNT and capable of causing a global catastrophe. Such events occur, on average of less than once in 100,000 years.

Within the next 100 or so years the probability is considerably lower than $10^{-5}$ in each year, as many of the potentially hazardous objects have been observed, and their orbits shown not to intersect with Earth. So while a naive estimation would suggest that the chance of an impact in the next 100 years is about 1 in 1000, the actual probability is rather less than that.

If you restrict to only comets, the probability is reduced further.

Answer 2

Your calculation of speeds etc is ok, and that kind of consideration might work for objects that essentially visit us once - e.g. very long period comets, or even comets that are falling into the Sun. Note that the Earth presents a somewhat bigger target than you suggest because of gravitational focusing, but this is a small effect for objects plunging towards the Sun from far away, since they will be travelling at relative speeds much greater than the Earth's escape velocity.

However, near-Earth orbiting objects pose a much greater hazard than your calculation suggests. For one thing, the closing speeds can be much lower, meaning the gravitational focusing effect can be much more significant. More importantly, because these objects are in $\sim 1$ year orbits, they get to have a go at hitting the Earth every time they cross the Earth's orbit.

We know that catastrophic impacts occur. It appears the last one was about 65 million years ago and one every 100 million years is probably a good ball park figure for how often these things are expected. If this were true, then your chances of being killed by a catastrophic impact are on a similar level to being killed in a commercial plane crash (if you fly on planes), but rather larger than being killed by something man-made falling out of the sky. How so? Well a catastrophic impact kills everyone; so for $\sim 10^{10}$ people on Earth, catastrophic impacts would kill (on average) 100 people/year. In a typical year of order 1000 people might die in commercial air crashes.

Your odds are an order of magnitude better than this though, because we are pretty certain ($\sim 90$%) that no very large ($>1$ km diameter) bits of rock will hit us in the next 100 years - so no catastrophic collisions are expected in that time. Smaller events are possible; and much more likely, because the population of smaller objects goes up as something like the square of the diameter. However, their mass and destructive power decreases as the cube of the diameter, so your individual risk from smaller events is I think lower than my original calculation for a "planet-killer".