Classical Dynamics of Particles and Systems · Problem 5-18

Calculus of Variations — Marion & Thornton

Physics — Mechanics · tidal-forces-moon-sun-ratio

Problem

Show that the ratio of maximum tidal heights due to the Moon and Sun is given by MmMs(REsD)3\frac{M_m}{M_s}\left(\frac{R_{Es}}{D}\right)^3 and that this value is 2.2. REsR_{Es} is the distance between the Sun and Earth, and MsM_s is the Sun's mass.
Intuitive explanation

Tidal forces arise from the difference in gravitational pull across Earth's diameter—not the average pull itself. The key insight is that tidal height is proportional to the gradient of gravitational force, which scales as M/r3M/r^3 for a body at distance rr. By comparing this r3r^3 dependence for the Moon (distance DD) versus the Sun (distance REsR_{Es}), you'll see why the Sun's greater distance overwhelms its greater mass, making the Moon's tides dominate despite the Sun being much more massive.

Intuitive explanation

Tidal forces arise from the difference in gravitational pull across Earth's diameter—not the average pull itself. The key insight is that tidal height is proportional to the gradient of gravitational force, which scales as M/r3M/r^3 for a body at distance rr. By comparing this r3r^3 dependence for the Moon (distance DD) versus the Sun (distance REsR_{Es}), you'll see why the Sun's greater distance overwhelms its greater mass, making the Moon's tides dominate despite the Sun being much more massive.

Related theory

Browse the problem library