Monthly Archives: April 2014

The probability of seeing a falling meteor while skydiving

This video is making the Internet rounds today, and it is absolutely astonishing.

A Norwegian skydiver, Anders Helstrup, was almost hit by a meteoroid hurling through the atmosphere.

I thought when I first saw this video, “WOW… what are the odds of that?”

I think that it’s possible to figure that out. This is a Fermi problem — there’s no right answer. But by putting together a series of guesses, one can come surprisingly close using basic estimation principles.

In 2012, the United States Parachute Association estimates there were 3.1 million skydives (which, in case you’re scoring at home, included 19 fatal accidents).

There were 49 fatal skydiving accidents worldwide, and 19 in the United States. Skydiving in the States is likely no more or less dangerous than elsewhere, and so it’s a reasonable assumption to conclude that since 40% of the fatalities happened in the US, then 40% of the jumps occurred here. So if there were 3.1 million jumps in the United States, there were (3.1 million) / 40% = 8 million jumps worldwide.

That was the easy part. Now, how to calculate the number of meteoroids that are falling on the Earth?

The one that passed “a couple of meters away” from Helstrup must have survived the atmosphere, at least in part. And per the meteorite hunter, it is likely a “bit bigger” than the one he held in his hand at 8:30 of the video. This appears to be about four inches (10 cm).

Using a power law given by Wikipedia, the number of rocks larger than 5 cm that impact Earth is approximated by N(>D) = 37D-2.7, where D is the diameter of interest (in meters). This formula’s only valid through about 5 cm, per the Wikipedia article. But this is a Fermi problem and we’re only approximating anyway, so it’s not a huge stretch to take it down to 1 cm. A centimeter is about as small as a flying rock would be and still be noticed. So N(>0.01m) = 37 × 0.01-2.7 ≈ 10 million impacts per year. (Sleep well tonight!)

Not only does a skydiver have to be near a falling meteoroid, she must be near a falling meteoroid at the same time it’s falling.

First, let’s look at physical distance. We’ll assume that the meteoroids fall uniformly over the Earth’s surface. But skydivers do not. They tend to only jump over land (go figure). Since land makes up about 29% of Earth’s surface, we take 29% of the impacts to even be eligible for skydiver proximity — down to 2.9 million per year.

I have no way of estimating how much of the Earth’s surface is actually likely to see skydivers. However, there are large, large swaths of Africa, Russia, Canada, and Australia that are so remote that there are few people and likely no skydivers. There’s also really no nice way to put this, but skydiving (requiring a private plane and all) is generally a sport limited to richer countries, so that will further restrict the places on Earth likely to see parachutists. With no proof at all I’ll estimate that about one-fifth of the Earth’s surface is skydive-able. That takes our impacts eligible for skydiver proximity from 2.9 million per year down to about 600,000 per year, or about 1600 per day.

Next let’s look at time. There are 600,000 falling rocks every year within range of a possible skydiver. How much time is spent in the air by rocks, and how much time by skydivers? Comparing these two numbers should give us a sense of the likelihood of the two being in the same place at the same time.

Most skydivers exit their plane at about 12,500 feet. This allows for a minute of freefall time, and about four minutes of canopy time, for a total of five minutes air time. Since there are about eight million jumps worldwide a year, skydivers spend approximately 40 million minutes in the air each year, or 2.4 billion seconds.

At an approximate terminal velocity of 200 miles per hour or so (and only counting the fact that we care about the lower 12,500 feet of its fall), a meteoroid is in the air for about 45 seconds. Multiply this by the 600,000 eligible meteoroids and we get 27 million seconds of annual meteoroid-in-the-air time.

27 million seconds of meteoroid time divided by 2.4 billion seconds of skydiver time means that on average, there is a meteoroid in the air for about 1% of the time that there’s a skydiver in the air. (Skydivers, sleep well tonight!)

But this is over the total skydive-able land of the Earth, which is about one-fifth of the Earth’s land area, or 6% of its surface. What are the odds that the meteoroid will be “within a couple of meters” of a skydiver?

Assuming that a meteoroid is likely to be anywhere within that 6% of Earth (about 12 million of the Earth’s 196.9 million square miles), and people are also likely to be anywhere within that 6% of Earth, then the odds of them being in the same square mile are one in 12 million.

But the odds of them being “within a couple of meters” of each other? Let’s assume that means the same ten square meters (a square 3.1 meters on a side). There are 2.6 million square meters in a square mile, so even if we could guarantee the skydiver and meteoroid were in the same square mile, the odds of the two being a couple of meters apart is about 0.0004% (four parts in a million).

So the combined odds are one in twelve million (8.3 × 10-8) times (4 × 10-6), or about 3.3 × 10-13 or 0.000 000 000 033%.

The probability of a skydiver and one-centimeter meteoroid intersecting in time is about 1%.

The probability of a skydiver and a one-centimeter meteoroid intersecting in space is about 0.000 000 000 033%.

Multiply these together, and the combined likelihood of what you saw in the video occurring is approximately 0.000 000 000 000 33%, or 3.3 &times 10-15. The probability is 300 trillion to 1.

Three hundred trillion to one.

Wow.

There are all kinds of approximations above, several of them I’m sure just outright wrong. There’s no real answer to this problem, only a very large ballpark estimate. I expect I may even be off a couple orders of magnitude here. But I’ll gladly accept feedback as to how to improve the estimate.