Model the tide the Sun raises on Earth's ocean

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

Relativity and Gravitation, Ch Tidal forces and the Newtonian tidal field

Problem

Model the tide the Sun raises on Earth's ocean. Take the Sun and the Moon as point masses, Earth as a sphere of radius

R_\oplus = 6.371\times10^{6}\ \mathrm{m}

whose centre falls freely in the Sun's field, and the ocean as a thin layer that responds only to the tidal field (Earth's rotation and the shape of coastlines are ignored). Data:

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

, Earth-Sun distance

d_\odot = 1.496\times10^{11}\ \mathrm{m}

M_{\rm Moon} = 7.35\times10^{22}\ \mathrm{kg}

, Earth-Moon distance

d_{\rm Moon} = 3.844\times10^{8}\ \mathrm{m}

. Use axes centred on Earth's centre with

x

pointing at the Sun and

y

z

transverse, and the relation

\Delta a_i = -\sum_j \mathcal{E}_{ij}\xi_j

. ((a)) Write the three non-zero components

\mathcal{E}_{xx}

\mathcal{E}_{yy}

\mathcal{E}_{zz}

of the Sun's tidal field at Earth's centre. ((b)) Find the relative acceleration of a water parcel at the sub-solar point,

\vec\xi = (R_\oplus,0,0)

, and of one at the antipodal point,

\vec\xi = (-R_\oplus,0,0)

, giving magnitude and direction relative to Earth's centre. Say what this implies for the number of solar high tides in a day. ((c)) Find the magnitude and direction of the relative acceleration of a parcel a quarter of the way round,

\vec\xi = (0,R_\oplus,0)

. ((d)) Compute the ratio of the Sun's tidal field strength at Earth to the Moon's. ((e)) Using that ratio, and taking the two tidal fields to add when Sun and Moon lie along the same line and to subtract when they are at right angles, compute the ratio of the spring tidal range to the neap tidal range.

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