Jupiter has mass

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames · Gravitation and central-force motion

Action and Symmetry, Ch Non-inertial frames from one Lagrangian

Problem

Jupiter has mass

M_{\rm J} = 1.898\times10^{27}\ \mathrm{kg}

. Its moon Europa orbits at a distance

d = 6.71\times10^{8}\ \mathrm{m}

from Jupiter's centre. Take Cartesian axes centred on Europa with the

x

-axis pointing from Europa towards Jupiter's centre, and treat Jupiter as a point mass. ((a)) Write down the tidal tensor

T_{ij} = G_{\rm N}M_{\rm J}(\delta_{ij}-3n_in_j)/d^3

of Jupiter's field evaluated at Europa's centre, giving

T_{xx}

T_{yy}

T_{zz}

and stating that the off-diagonal components vanish in these axes. ((b)) Verify that the trace of

T_{ij}

is zero.

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