Earthquake maths: A Whole Log Of Shaking

(Photo courtesy of UN Development Program.)

We've had a lot of noticeable earthquakes around the world in the last year or so - Haiti, Chile, New Zealand. Probably no more than usual, except that we notice them more - but that's a topic for another day.

Instead, we're going to talk about how we measure the size of an earthquake. You'll quite commonly hear that the Haiti earthquake measured 7.0 on the Richter Scale, but that's not quite true[1] - the Richter Scale has gone the same way as the furlong and the acre and been replaced by a more scientific measurement, the moment magnitude.

Fortunately, for medium-size earthquakes (3.0 to 7.0 on either scale), the two numbers are pretty similar - as I understand it, it's not possible to gather the data for the Richter scale above about 6.8, but I look forward to being corrected by seismologists.

What does the number mean?

The moment magnitude is a measure of how much mechanical work was done by the quake. Now, work in this context has a specific, physical meaning - it's not a measure of how the earthquake gave a sharp intake of breath and charged you £400 to replace your brakes. Instead, it's closely related to how much energy the earthquake released. And for that, we're going to use logs.

No! Not the logs! Anything but the logs!

Oh, grow up. The moment magnitude scale is a logarithmic scale. That means a 6.0 magnitude earthquake isn't twice as strong as one with 3.0 magnitude. Instead, you need to apply a formula: $relative~strength = 10^{1.5(M-m)}$, where $M$ is the big magnitude and m the small one.

So, for that example, we'd do $10^{1.5\times 3} = 10^{4.5}$ = about 32,000 times bigger - slightly more than double, right[2]?

If we wanted to compare the 7.0-magnitude quake in Haiti with the 8.8-magnitude Chile one, we find that the Chilean earthquake was $10^{1.5\times 1.8} = 500$ times stronger. (It did a lot less damage, both human and material, as it was offshore and Chile's infrastructure was better-equipped to deal with a disaster). And to compare to the 2008 Market Rasen earthquake, which had the devastating effect of waking me up[3] with a magnitude of 5.2: Chile's quake was $10^{1.5 \times 3.6}$ stronger - a factor of about 250,000.

Market Rasen released about 4 gigajoules of energy - enought to run a lightbulb for about 16 months. Chile's quake, then - a quarter of a million times bigger - would have released a million gigajoules, enough to run the lightbulb for 300,000 years, give or take - or a city of a million people for about three weeks.

So, there you go. A real-life application of logarithms and powers.


To donate to the Disaster Emergency Committee, who are working to rebuild areas hit by catastrophic events - including the Haiti earthquake and Pakistan floods - please click here


[1] This was news to me, too. Thank you, wikipedia!
[2] Interestingly, mechanics use a similar concept when telling you how long it'll take to fix your car, and how much it'll cost.
[3] I'm quite proud that I woke up, thought 'oh! an earthquake!' and went back to sleep.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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