The Gutenberg-Richter Energy Relation Answers in Ergs
The first equation most seismology courses hand out is log₁₀ E = 11.8 + 1.5 M, and the E it returns is in ergs, not joules. That is not an oversight: Gutenberg and Richter were working in the CGS system, and the constant 11.8 is calibrated to it. Put M = 5.0 into the formula and you get 10¹⁹·³ = 2.0e19 erg — a number that means nothing until it becomes 2.0e12 J and can be set beside a power station or a quarry blast.
What the Formula Is and Is Not Telling You
The 11.8 Is a CGS Constant
Each Step Is 31.6 Times, Not 10
Moment and Radiated Energy Measure Different Things
Richter Saturates, Moment Magnitude Does Not
Working a Magnitude Through to a Number You Can Picture
The formula, the unit step and the comparison are three separate operations. Doing them in that order keeps the exponents under control and makes an answer easy to check against the table below.
Evaluate the relation first
Compute 11.8 + 1.5 M, then raise ten to that power. For M = 7.0 the exponent is 22.3 and the answer is 2.0e22 erg. Nothing about this step involves joules yet — resist converting the magnitude itself, which is dimensionless.
Point the pair from erg to joule
Since the textbook already produced an erg value, press the swap arrows so ergs are the input side and joules the answer. Typing works in whichever direction the pair is set, so nothing is lost if you decide to go back.
Enter the exponent form as it stands
Values this large are returned in scientific notation automatically rather than as twenty-odd digits, and a decimal comma is accepted alongside a point — useful when the worked example came from a European course pack.
Reach for a comparison unit
Switch the joule side to kWh or MJ from the searchable menu to see the answer as something billable: that M 7.0 event is 5.54e8 kWh, roughly what a one-gigawatt station generates in 23 days.
Earthquake Magnitudes and the Energy They Radiate
Every row below comes straight from log₁₀ E = 11.8 + 1.5 M, converted at 1e7 erg per joule. Reading down the table shows the 31.6-fold step far more vividly than the formula does.
| Magnitude | Radiated energy (erg) | In joules | Comparable to |
|---|---|---|---|
| M 2.0 | 6.31e14 | 6.31e7 | About 17.5 kWh, most of a day of household electricity |
| M 3.0 | 2.00e16 | 2.00e9 | Roughly a single lightning flash |
| M 4.0 | 6.31e17 | 6.31e10 | A US household's electricity for about 20 months |
| M 5.0 | 2.00e19 | 2.00e12 | A 550 MW station running for one hour |
| M 6.0 | 6.31e20 | 6.31e13 | The 15-kilotonne Hiroshima blast |
| M 7.0 | 2.00e22 | 2.00e15 | A 1 GW plant running about 23 days |
| M 8.0 | 6.31e23 | 6.31e16 | Some five hours of world electricity generation |
| M 9.0 | 2.00e25 | 2.00e18 | Close to a week of world electricity generation |
The last two rows explain why great earthquakes dominate the global seismic energy budget: a single M 9 releases more than thirty thousand M 6 events combined, so a decade's worth of moderate shaking barely registers against one subduction-zone rupture.
What the Converter Adds to a Magnitude Calculation
Magnitude-9 Totals Move to Exponent Form
Once a result passes 1e10 it is shown in scientific notation, so an M 9 figure reads as 2.00e25 rather than a twenty-six-digit string. Counting zeros is exactly where magnitude arithmetic normally goes wrong.
kWh and Megajoules for the Comparison Column
The searchable menu on either side reaches kWh, MJ, GJ and kcal, which is how an abstract exponent turns into a household bill or a plant's output. The comparison lands without a second calculation.
Start in Ergs When the Textbook Already Did
The swap arrows put the erg field first, matching the direction a worked example actually runs. Reverse it again to restate a modern SI energy determination in the CGS form an older paper uses.
Plain Digits for the Problem Set
The copy control above each field takes the value with no unit attached, so an exponent lands cleanly in a written answer or a spreadsheet column instead of arriving with stray characters that have to be deleted.
Magnitude, Moment and Energy Questions from a Seismology Course
Why is log E = 11.8 + 1.5M written for ergs rather than joules?
Because the relation dates from the 1950s, when seismology worked entirely in CGS: displacements in centimetres, moments in dyne-centimetres, energies in ergs. The intercept 11.8 is a fitted constant tied to that choice of unit. If you prefer SI throughout, the same expression becomes log₁₀ E(J) = 4.8 + 1.5 M, because dividing by 1e7 subtracts seven from the exponent. Both forms give identical physics; only the printed constant changes.
Where does the factor of 31.6 per magnitude step come from?
Straight from the 1.5 coefficient. Adding one to M adds 1.5 to the exponent, and 10¹·⁵ is 31.62. The confusion arises because magnitude was originally defined from ground-motion amplitude, where one step really is a factor of ten — but a tenfold larger wave carries about thirty-two times the energy, since energy scales with both amplitude and duration. So the amplitude answer and the energy answer are both correct about different quantities.
Is seismic moment in dyne-centimetres the same thing as radiated energy?
No, although both are energies dimensionally and both are traditionally quoted in CGS. Moment M₀ describes the size of the rupture — shear modulus times fault area times average slip — and 1 dyne·cm equals 1e-7 N·m, the same factor as erg to joule. Radiated energy is only the fraction that escapes as seismic waves, around 5e-5 of the moment for a typical event; the rest goes into fracture, friction and heat on the fault plane. An Mw 7.0 has M₀ near 3.55e26 dyne·cm but radiates only about 2.0e22 erg.
Why did moment magnitude replace the Richter scale?
Because the original scale saturates. Local magnitude was defined from the peak amplitude on a specific short-period instrument in southern California, and above roughly M 7 that instrument stops distinguishing larger ruptures — a 300 km fault and a 1 000 km fault produce similar readings. Moment magnitude, Mw = (2/3) log₁₀ M₀ − 10.7 with M₀ in dyne-centimetres, is built from the rupture geometry itself, so it stays meaningful all the way to the largest events. The 1.5 coefficient in the energy relation is the same 3/2 that appears inverted in that definition.
What everyday energy is a magnitude 6 earthquake comparable to?
6.31e13 J, which is 1.75e7 kWh — the annual electricity of well over a thousand average homes, or a gigawatt station's output for about 17 hours. It also sits within one per cent of the 15-kilotonne Hiroshima yield, which is why that comparison is so often reached for. What the equivalence hides is the delivery: an earthquake releases its energy along tens of kilometres of fault over many seconds, so nothing near the surface experiences anything like a bomb.
No comments yet. Be the first to comment!