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Unformatted text preview: 1 Exercises on Gravitational Waves 1. Multipolar expansion of the gravitational field Show that the gravitational field written in a multipolar expansion as h ∼ G c 2 I r + G c 3 ˙ I 1 r + G c 4 ¨ I 2 r + ··· + G c 4 ˙ J 1 r + G c 5 ¨ J 2 r + ··· is dimensionless, where r is the distance from the source, I , I 1 and I 2 are the mass monopole, dipole and quadrupole, respectively, and J 1 , J 2 are the current dipole and quadrupole, respectively. Assuming that the gravitational field is proportional to G/r (where r is the distance between the observer and the source, and G is the Newton constant), and that each term of the expansion depends only on c and on derivatives of the multipole mass-moments I L and current-moments J L , show that the multipolar expansion is unique. 2. Orders of magnitude Use the leading, non-zero term in the multipolar expansion of the gravitational field, i.e., the quadrupolar term G ¨ I 2 / ( c 4 r ), to estimate the amplitude of gravitational waves produced from the following earth-based events: • A meteorite having diameter of 2 km and hitting the ground at a speed of 25 km / sec; • A big chunk of piezoelectric material driven to oscillation at a frequency of 100 MHz.A big chunk of piezoelectric material driven to oscillation at a frequency of 100 MHz....
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- Fall '08