Chapter 13399Additional ProblemsP13.47Let mrepresent the mass of the spacecraft, rEthe radius of the Earth’s orbit, and xthe distance fromEarth to the spacecraft.The Sun exerts on the spacecraft a radial inward force of FGM mrxssE=−bg2while the Earth exerts on it a radial outward force of FGM mxEE=2The net force on the spacecraft must produce the correct centripetal acceleration for it to have anorbital period of 1.000 year.Thus,FFGM mGM mxmvmTSESEEEEE−=−−=−−LNMMOQPP22222πwhich reduces toGMGMxTSEEE−−224.(1)Cleared of fractions, this equation would contain powers of xranging from the fifth to the zeroth.We do not solve it algebraically. We may test the assertion that xis between 1 47 109.×m and148 109.×m by substituting both of these as trial solutions, along with the following data:MS=×1991 1030. kg,ME5983 1024kg,rE1496 1011.m, and T==×10003156 107..
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