Issue 022 - Earthquakes - Logarithmic scaling
How much energy did the Japan earthquake release?
Reuters reported that a major earthquake struck near Kumamoto in southwestern Japan on July 28, 2026. Japan's meteorological agency reported magnitude 7.1, while the USGS later estimated 6.8. Use the 7.1 value for the main estimate, then check how much the answer changes at 6.8.
The problem
Estimate the energy released by a magnitude 7.1 earthquake, using a magnitude 6 earthquake as 6 x 10^13 joules.
Each increase of one magnitude represents about 32 times more energy. Convert the result into a familiar comparison, such as gasoline, household-years of electricity, or the energy of the 20 reentering Starlink satellites from the earlier problem.
Then estimate how many magnitude 5.1 earthquakes would release the same combined energy, and do a sensitivity check for the USGS estimate of magnitude 6.8.
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I relied heavily on the given information.
If each change in magnitude of 1.0 corresponds to an increase or decrease of 32x the prior magnitude value, then each change in magnitude of 0.1 should be about 3.2x difference.
Going from a 6.0 to a 7.1 should correspond to a 35x increase, roughly. If a magnitude 6.0 is equivalent to a release of 6 x 10^13 J, then multiplying by 35 should give the energy release in a 7.1 magnitude quake:
energy ~= 6 x 10^13 J x 35
~= 2.1 x 10^15 JIf a liter of gasoline contains 4.5 x 10^7 J, dividing that into the 7.1 earthquake energy release gives about 46 million liters of gasoline.
If a household uses about 1.1 MWh of energy monthly, then:
annual household electricity ~= 1.1 MWh/month x 12 months
~= 13.2 MWh/year
13.2 MWh/year ~= 13,200 kWh/year
~= 13,200 x 3,600,000 J
~= 4.7 x 10^10 J/yearDividing that into the 7.1 earthquake energy estimate gives about the same energy used by 44,000 houses per year.
A 5.1 magnitude earthquake is about 1/29th as powerful as the 6.0, so I estimated it as around 2 x 10^12 J.
If it was a 6.8 magnitude earthquake instead of a 7.1, I estimated that as about 26x as powerful as a 6.0:
6.8 estimate ~= 6 x 10^13 J x 26
~= 1.6 x 10^15 JThat would be a decrease of about 5 x 10^14 J from my 7.1 estimate.
Calibration Score
Matt's Calibration Score: 55 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 20/30. The one-magnitude 32x peg was remembered, but the fractional step was mishandled.
Model: 0/30. Using 3.2x for each tenth of magnitude is not the right logarithmic model.
Math: 5/10. The fractional-magnitude arithmetic mistake mattered conceptually.
Result: 30/30. The final energy estimate was still very close because the mistake did not move this specific comparison much.
Grounding facts
Total earthquake energy is not the same thing as damage. USGS explains that magnitude is one value for the earthquake, while shaking intensity varies from place to place based on distance, surface material, rupture direction, local geology, and other factors. Depth matters too: a deeper quake can produce less severe local shaking than a shallower quake of similar magnitude.
So the energy estimate explains one part of the event's scale, but not the full human impact. Damage also depends on how close people and structures are to the rupture, how the ground transmits shaking, whether landslides, fire, or tsunami occur, and how buildings, bridges, roads, utilities, and emergency systems are built.
After checking sources
Check and recalibrate
The main correction is the fractional-magnitude step. A one-magnitude increase is about 32x energy, but a tenth of a magnitude is not 3.2x. The shortcut is:
energy ratio ~= 32^dM
~= 10^(1.5 x dM)For a magnitude 7.1 earthquake compared with the magnitude 6 anchor, the magnitude difference is 1.1:
ratio ~= 32^1.1
~= 10^(1.5 x 1.1)
~= 10^1.65
~= 45
energy ~= 6 x 10^13 J x 45
~= 2.7 x 10^15 JMatt's first estimate of 2.1 x 10^15 J was low by only about 30%, which is very close for a Fermi problem. The method mistake mattered, but the final answer stayed in the same order of magnitude.
For gasoline, use the site's existing EIA gasoline peg of about 1.3 x 10^8 J per gallon, or roughly 3.4 x 10^7 J per liter:
gasoline gallons ~= (2.7 x 10^15 J) / (1.3 x 10^8 J/gallon)
~= 2.1 x 10^7 gallons
gasoline liters ~= 2.1 x 10^7 gallons x 3.8 L/gallon
~= 8 x 10^7 litersSo the magnitude 7.1 estimate is roughly 20 million gallons, or about 80 million liters, of gasoline energy. If you use Matt's 4.5 x 10^7 J/L gasoline peg instead, the comparison is about 60 million liters. Either way, the scale is tens of millions of liters.
For household electricity, use about 10,000 to 13,000 kWh per household-year:
household-year energy ~= 4 x 10^10 J/year
household-years ~= (2.7 x 10^15 J) / (4 x 10^10 J/year)
~= 7 x 10^4 household-yearsThat is roughly 60,000 to 70,000 household-years of electricity, depending on which household-use peg you choose.
Compared with the earlier Starlink reentry estimate of about 10^12 J:
Starlink reentry equivalents ~= (2.7 x 10^15 J) / (1 x 10^12 J)
~= 2.7 x 10^3So this quake is about 3,000 of those 20-satellite reentry-energy events.
For magnitude 5.1 earthquakes, compare directly with 7.1. The difference is exactly two magnitude units:
same-energy M5.1 quakes ~= 32^2
~= 1,024
~= about 1,000You can also reach the same answer by estimating one M5.1 quake from the M6 anchor:
M5.1 is 0.9 below M6
M5.1 energy ~= 6 x 10^13 J / 32^0.9
~= 6 x 10^13 J / 23
~= 3 x 10^12 J
2.7 x 10^15 J / 3 x 10^12 J
~= 1,000For the USGS 6.8 sensitivity check, the magnitude difference from M6 is 0.8:
ratio ~= 32^0.8
~= 10^(1.5 x 0.8)
~= 10^1.2
~= 16
M6.8 energy ~= 6 x 10^13 J x 16
~= 1 x 10^15 JThe 6.8 estimate is about 1 x 10^15 J. That is about 2.8 times smaller than the 7.1 estimate. The difference, roughly 1.7 x 10^15 J, is itself an enormous same-order-of-magnitude energy difference.
Post-check reflection
Matt's reflection
I did not do a great job estimating the difference in energy as the magnitude changed. That 10^(1.5 x delta magnitude) formula seems like a valuable one to keep track of.
Still, my answers were pretty close, within an order of magnitude.
As for the news item, the scale of the energy release is so enormous that it is still difficult to conceptualize. I know we put it in terms of liters of gasoline and household annual energy consumption, but even so it is still so large that I do not know what to do with these numbers. The difference between a 7.1 and a 6.8 magnitude earthquake is on the same order of magnitude as both of them.
Recommended memory peg
Remember one earthquake magnitude step is about 32x energy, and the flexible shortcut is energy ratio ~= 10^(1.5 x magnitude difference). A 0.1 magnitude step is about 1.4x, not 3.2x.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.