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2. Multipole Expansion in Cartesian Coordinates The gravitational potential is given by

2. Multipole Expansion in Cartesian Coordinates The gravitational potential is given by

Applications and Investigations in Earth Science (9th Edition)
Applications and Investigations in Earth Science (9th Edition)
9th Edition
ISBN:9780134746241
Author:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Publisher:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Chapter1: The Study Of Minerals
Section: Chapter Questions
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2. Multipole Expansion in Cartesian Coordinates The gravitational potential is given by * (7) = -G [ d²r _L(F") 『ーデ Assume that the mass distribution p(F) is localized so that p(F") × 0 if r'> rmaz- Approximate the potential for the case r> rmax. a) Proof that the first term in a multipole expansion has the form GM Po (F) b) Derive the term for the second order contribution (dipole moment) and express it by using the center of mass R. c) What is the gravitational dipole moment in the center of mass references frame?
Transcribed Image Text:2. Multipole Expansion in Cartesian Coordinates The gravitational potential is given by * (7) = -G [ d²r _L(F") 『ーデ Assume that the mass distribution p(F) is localized so that p(F") × 0 if r'> rmaz- Approximate the potential for the case r> rmax. a) Proof that the first term in a multipole expansion has the form GM Po (F) b) Derive the term for the second order contribution (dipole moment) and express it by using the center of mass R. c) What is the gravitational dipole moment in the center of mass references frame?
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