The Sun has mass and Mercury orbits at a mean distance from the Sun's centre

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

The Sun has mass

M_\odot = 1.989\times10^{30}\ \mathrm{kg}

and Mercury orbits at a mean distance

d = 5.79\times10^{10}\ \mathrm{m}

from the Sun's centre. Treat the Sun as a point mass. Take Cartesian axes centred on Mercury's centre such that the unit vector towards the Sun is

\mathbf n = (\cos45^\circ,\sin45^\circ,0)

, i.e. tilted at

45^\circ

to the

x

-axis in the

xy

-plane. ((a)) Find all the (generally nonzero) components

T_{xx}

T_{yy}

T_{zz}

T_{xy}

of the Sun's tidal tensor at Mercury's centre. ((b)) Verify that the trace vanishes.

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