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Math 111: Logarithm Scales

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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 The Richter scale has since been replaced by the Moment Magnitude scale, which is instead based on an estimate of the seismic energy released by the earthquake.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 M = log(M ) − 10:7 w 3 0 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? 2 M = log(2 × 1025) − 10:7 ≈ 6:17 w 3 Example 2: The 2010 Haiti earthquake had a moment magnitude of 7:0. About how much seismic energy was released? 2 3 (7:0+10:7) 26 7:0 = log(M ) − 10:7 ) M = 10 2 ≈ 3:55 × 10 dyne · cm 3 0 0 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 3 (9:5+10:7) 30 9:5 = log(M ) − 10:7 ) M = 10 2 ≈ 2:00 × 10 dyne · cm 3 0 0 ...and divide the two. 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(V ) − 10:7 − log(H ) − 10:7 = log(V ) − log(H ) 3 M0 3 M0 3 M0 M0 2 VM0 VM0 3 (2:5) ) 2:5 = log ) = 10 2 ≈ 5623 3 HM0 HM0 ...and again we have that the Valdivia earthquake released more than 5600 times as much energy as the Haitian earthquake. 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 p f f m = − 5 100 log ≈ −2:5 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. Example 4: What is the stellar magnitude of Vega? fV mvega = −2:5 log = −2:5 log(1) = 0 fV In fact, a magnitude 0 star is quite bright! Example 5: What would be the stellar magnitude of a star which is one tenth as bright as Vega? 0:1f m = −2:5 log V = −2:5 log(0:1) = +2:5 fV A star with a positive stellar magnitude is dimmer than Vega. Example 6: The stellar magnitude of the bright star Sirius is -1.47. Which star is brighter, Sirius or Vega? How much brighter? Sirius is brighter than Vega. It is the brightest star in the sky (besides the Sun). fsirius msirius = −1:47 = −2:5 log fV log fsirius = −1:47 = 0:588 fV −2:5 fsirius log 0:588 10 fV = 10 ) fsirius ≈ 3:87fV So Sirius is almost four times brighter than Vega. Example 7: 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? Betelgeuse is brighter. 0:45 − 1:26 = −2:5 log fbetel − −2:5 log fdeneb fV fV −0:81 = −2:5 log fbetel − log fdeneb fV fV fbetel −0:81 = log fV −2:5 fdeneb fV 0:324 = log fbetel · fV fV fdeneb fbetel 0:324 log 10 = 10 fdeneb ) fbetel ≈ 2:11fdeneb So Betelgeuse is a little more than twice as bright as Deneb. Math 111 Homework: Logarithm Scales 1. Use the Moment Magnitude scale to answer the following questions about earthquakes. (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. Use the apparent stellar magnitude scale to answer the following questions about stars. (a) How much brighter is the star Canopus (m = −0:72) than Vega? (b) Which is brighter, Aldebaran (m = 0:85) or Fomalhaut (m = 1:16)? How much brighter?.
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