Energy Ratio Between Two Magnitudes

Take the energy relation, write it for two earthquakes, and subtract. The calibration constant vanishes and you are left with $E_2/E_1 = 10^{1.5\,\Delta M}$ — a pure statement about the scale, independent of any era's instruments and as exact as the definition of magnitude itself. This is the single most useful fact in this whole shard.

Two numbers are worth committing to memory. One magnitude unit is $10^{1.5} = 31.6$ times the energy, usually rounded to "about 32". Two magnitude units is $10^3 = 1000$ times, exactly. The second is the cleaner mnemonic and the easier one to check: a magnitude 8 releases a thousand times the energy of a magnitude 6, and a million times that of a magnitude 4.

Now the confusion this page exists to clear up. There are two "how much bigger" numbers attached to magnitude and they are not the same. The AMPLITUDE recorded on a seismogram goes up by a factor of ten per magnitude unit — that is the original definition of the scale, a logarithm of amplitude. The ENERGY goes up by a factor of 31.6. Ten times the wiggle, thirty-two times the energy, and the discrepancy is not an inconsistency: energy in a wave goes as amplitude squared times duration, and larger earthquakes shake for longer as well as harder, which is where the extra factor comes from. Quoting one number where the other belongs is the commonest error in popular accounts of earthquakes.

The practical consequence is that the phrase "only one point higher" should never be spoken. The 2011 Tohoku earthquake at Mw 9.1 released roughly 32 times the energy of a magnitude 8.1, and about a thousand times that of the magnitude 7.1 that would already have been a serious disaster. Run the same arithmetic across a whole region and you get the point the Gutenberg–Richter line makes: with $b \approx 1$ there are ten times as many magnitude 6s as magnitude 7s, but each 7 carries 32 times the energy, so the largest events dominate the total by a factor of about three per magnitude step, all the way up. Everything smaller is, energetically, rounding error.