Download Math 111: Logarithm Scales

Math 111: Logarithm Scales
Earthquakes used to be measured according to a logarithmic scale called the Richter
scale which was based on the amplitude of the vibrations from the earthquake. The Richter
scale has since been replaced by the Moment Magnitude scale, which is instead based on an
estimate of the seismic energy released by the earthquake.
The formula is
2
Mw = log(M0 ) − 10.7
3
where M0 is the seismic moment of the earthquake, measured in dyne-cm (a unit of energy).
Example 1: The Long Beach earthquake of 1933 released 2 × 1025 dyne-cm of seismic
energy. What was the moment magnitude of this earthquake?
Mw =
2
log(2 × 1025 ) − 10.7 ≈ 6.17
3
Example 2: The 2010 Haiti earthquake had a moment magnitude of 7.0. About how
much seismic energy was released?
7.0 =
2
log(M0 ) − 10.7
3
⇒
3
M0 = 10 2 (7.0+10.7) ≈ 3.55 × 1026 dyne · cm
Example 3: The strongest earthquake since 1900 was the 1960 Valdivia earthquake in
Chili. It had a moment magnitude of 9.5. How much stronger was the Valdivia earthquake
than the Haitian earthquake of 2010?
Well, we could just find the seismic moment of the Valdivia earthquake and compare...
2
log(M0 ) − 10.7
3
...and divide the two.
9.5 =
⇒
3
M0 = 10 2 (9.5+10.7) ≈ 2.00 × 1030 dyne · cm
2 × 1030
≈ 5634
3.55 × 1026
That means the Valdivia earthquake released more than 5600 times as much energy as the
Haitian earthquake.
A simpler way to do this is to subtract the two magnitudes and simplify.
2
2
2
9.5 − 7.0 =
log(VM 0 ) − 10.7 −
log(HM 0 ) − 10.7 =
log(VM 0 ) − log(HM 0 )
3
3
3
3
2
VM 0
VM 0
⇒ 2.5 = log
⇒
= 10 2 (2.5) ≈ 5623
3
HM 0
HM 0
...and again we have that the Valdivia earthquake released more than 5600 times as much
energy as the Haitian earthquake.
Math 111 Homework: Logarithm Scales
1. Recall that the Moment Magnitude scale for measuring the size of an earthquake is
defined to be:
Mw =
2
log(M0 ) − 10.7
3
where M0 is a very rough estimate of the seismic energy of an earthquake (in dynes-cm).
(a) The Kern County earthquake of 1952 released about 2 × 1027 dyne-cm of seismic
energy. What was the moment magnitude of the Kern County earthquake?
(b) The El Centro earthquake of 1940 had a moment magnitude of 7.0. About how
much seismic energy was released?
(c) How much more seismic energy was released by the Sumatra earthquake of 2004
(Mw = 9.3) than the San Francisco earthquake of 1906 (Mw = 7.9)?
2. The observed brightness of a star is formally its energy flux or the power per unit area
radiated by the star. If measured in a unit like watts per cubic meter, this is a very
small number which also varies by many orders of magnitude from star to star. For a
more manageable parameter for the brightness of a star, a logarithmic scale called the
apparent stellar magnitude is used. It is defined to be
√
f
f
5
≈ −2.5 log
m = − 100 log
fV
fV
where f is the energy flux of the star and fV is a reference energy flux which for
historical reasons is approximately the energy flux of the bright northern star Vega.
(a) What is the stellar magnitude of Vega?
(b) What would be the stellar magnitude of a star which is one tenth as bright as
Vega?
(c) The stellar magnitude of the bright star Sirius is -1.47. Which star is brighter,
Sirius or Vega? How much brighter?
(d) The star Betelgeuse has a magnitude of 0.45 while the star Deneb has a magnitude
of 1.26. Which star is brighter? How much brighter is it?