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Sol10 - Solution to PHYS675 HW 10(Fall 2010 December 2 2010...

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Unformatted text preview: Solution to PHYS675 HW 10 (Fall 2010) December 2, 2010 1 Multipolar expansion of the gravitational eld 1.1 Prove that h is dimensionless We know the dimensions for the following quantities: [ G ] = L 3 MT 2 [ I n ] = ML n [ J n ] = ML n /T [ c ] = L/T [ r ] = L Then one can directly verify that all terms in the expansion are dimension- less. 1.2 Uniqueness of multipolar expansion Under the assumption that the gravitational eld is proportional to G/r ,which has the dimension L 2 MT 2 , we conclude that the rest of each term must have dimension MT 2 /L 2 , and therefore can only be linear in either I n or J n (or their time derivatives). The only physical dimensional parameter left is c, which helps to remedy the dimensions. In this way we have proven the uniqueness of the multipolar expansion, up to coe cients before each term. 1 2 Orders of magnitude Since I 2 ≈ ML 2 ,we have ¨ I 2 = Mv 2 ,in which v is the characteristic velocity of the source. Therefore h + ,h × ≈ G c 4 r Mv 2 Now we estimate the amplitude of the gravitational waves emitted by the following two events: 2.1 Meteorite with a diameter of 2km, hitting the ground at a speed of 25km/sec. According to the diameter, the volume of the meteorite is V ≈ 8 × 10 9 m 3 ....
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Sol10 - Solution to PHYS675 HW 10(Fall 2010 December 2 2010...

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