A spacecraft of length is held with its long axis pointing exactly along the radius…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Newtonian gravitation as a field theory

The Geometry of Gravity, Ch The tidal field: what free fall cannot remove

Problem

A spacecraft of length ℓ=5.00 m\ell = 5.00\ \mathrm{m} is held with its long axis pointing exactly along the radius from the Sun, at Mercury's mean orbital distance r=5.79×1010 mr = 5.79\times10^{10}\ \mathrm{m}. Treat the Sun as a point mass M⊙=1.989×1030 kgM_\odot = 1.989\times10^{30}\ \mathrm{kg} and use G=6.674×10−11 N m2 kg−2G = 6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}. Ignore the spacecraft's own gravity and any non-gravitational forces. ((a)) Write Eij=∂i∂jΦ\mathcal{E}_{ij} = \partial_i\partial_j\Phi in axes aligned with the radius (one radial, two transverse) as a diagonal matrix with numerical entries in s−2\mathrm{s^{-2}}, and confirm that its trace vanishes. ((b)) Find the tidal acceleration −EijSj-\mathcal{E}_{ij}S^j between the two ends of the spacecraft.

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