Thursday, September 22, 2011

Calculating the (Unlikely) Odds of NASA Satellite Casualties

How did NASA come up with the 1 in 3200 chance of anyone on Earth being hit by pieces of the Upper Atmosphere Research satellite (UARS) due to crash on Friday?



We contacted NASA yesterday to ask them what goes into their estimate of the risks that incoming satellites present. In short, the answer we received from NASA's Nick Johnson is "A complex computer program called ORSAT is used," to do the calculation. (The complete NASA reply is available at the end of this post.)

That's cool, but computer output is only as good as the input and programming. In my experience, it's always a good idea to get out a pencil and paper to make sure the computer's answer makes sense. So I thought I'd try to estimate the odds UARS hitting anyone all by myself. Here goes . . .

One thing we know, thanks to NASA press officers, is that UARS will hit somewhere between 57 north and 57 south latitude. If you happen to have a globe handy, you will see that the impact zone includes most of the populated areas of the planet. Northern Europe, as well as the upper parts of Siberia, China and Canada are safe, as it Antarctica, but the places where the vast majority of the planet's 7 billion people live are between 57 north and 57 south latitude. By just eyeballing the globe, I'd say that roughly 4/5 of the planet's surface falls in that region as well.

About three quarters of the Earth is covered in water, and I'm going to assume for simplicity that there are so few people on the water at any given moment that we can just ignore all the wet portions. That means there's only a 1 in 4 chance of the satellite hitting places where people might be.

I'm going to count it as a hit if the satellite gets within one square meter of any person, which means there's about 7 billion square meters of human that we have to worry about.

Now, there's approximately 150,000,000 square kilometers of dry land on Earth. Because 4/5 of that land is in the danger zone, we only have about 120,000,000 square kilometers at risk.

We have to convert the area humans cover from square meters to square kilometers to figure out the odds that a person will be hit.

(7,000,000,000 square meters)/(1,000,000 square meters per square kilometer) = 7,000 square kilometers of human

To figure out the chances of the satellite landing on a bit of land with a human on it, we just have to divide the total land area into the area of land covered in people.

(7,000 square kilometers of people)/(120,000,000 square kilometers of land) = about 1/17,000

So there's roughly a 1 in 17,000 chance that the satellite will hit a person, if it hits land at all. But there's only a 1 in 4 chance of it hitting land, so divide 1/17,000 by four to get 1/68,000 that the satellite will hit a person.

One thing I didn't include yet is that the satellite will break up into several pieces as it reenters the atmosphere. If you guess that it will break into about 10 pieces, the odds get worse by a factor of 10 (because the same calculation applies to each piece) to about 1 in 6800, which is only about twice the NASA estimate of 1 in 3200.

There you have it, the roughest of rough estimates agrees with NASA's computer calculations. I'm sure that makes everyone more comfortable with the risks we face tomorrow.

Update: I just learned that NASA expects the satellite to break into about two dozen pieces. If I take that into account in my calculation, my estimate changes to 1 in 2830, which is shockingly close to the NASA estimate. (In fact, I wouldn't have believed it if I hadn't done the calculation from scratch myself.)

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If you want to know more about how NASA did the calculations, here are Nick Johnson's answers to our questions.

1. How did you/NASA arrive at the "estimated human casualty risk" of 1/3,200 people? Was this done through computer modeling or a back-of-the-envelope type calculation?

A complex computer program called ORSAT is used. A lower fidelity routine can be found in NASA's DAS software, which can be downloaded from http://www.orbitaldebris.jsc.nasa.gov/mitigate/das.html.


2. What are the assumptions that went into the calculation?

The initial state of the vehicle at an altitude of 122 km, e.g., flight trajectory and temperature, are needed for an assessment. Breakup altitude (typically 78 km) must also be input. However, these values are run parametrically if there is any uncertainty which might affect the answer. Note that most satellite components will survive or will demise under a wide variety of initial conditions. Only those components which appear to be borderline, i.e., heat of ablation is not quite reached or just barely reached, normally require further evaluation.


3. How many assumptions were made?

See above.


4. Is there redundancy in the risk assessment and if so, how was it accomplished? (For example, if a computer model calculated the risk, did a human do a paper and pencil calculation to make sure the number was reasonable?)

Paper and pencil is not an option for a complex spacecraft. The ORSAT model has been verified and validated in a number of different ways.


5. Could you explicitly show your calculations to us?

A summary of the assessment is provided in the attachment, a paper delivered at the 2002 World Space Congress. (see the abstract of the paper here, or email us for a full copy)

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