Past exam of the physics B course of the University of Cambridge / 2018 / Paper 1 / Section C / C11

The differential gravitational acceleration responsible for a tidal force is the difference between accelerations at separated points of a body. Relative to a freely falling reference point $\mathbf{R}$, a small offset $\mathbf{\xi }$ experiencesThe tidal tensor has units ${\mathrm{s}}^{-2}$; the vector called $\mathbf{T}(\mathbf{\xi })$ in the source is the resulting differential acceleration, in $\mathrm{m}{\mathrm{s}}^{-2}$. An actual differential force is this vector multiplied by the test mass. Measurable effects include relative motion of separated freely falling masses, radial stretching, transverse squeezing, deformation and, when self-gravity or strength is insufficient, disruption. A spatially uniform acceleration alone produces no tidal deformation.

For a point mass, $\mathbf{g}(\mathbf{R})=-GM\mathbf{R}/R^3$. Along the outward radial direction, $g_R=-GM/R^2$ andThus a small radial displacement $\mathbf{a}=a\hat{\mathbf{R}}$ has differential accelerationThe farther point falls inward less strongly, so its acceleration relative to the reference point points outward. The formula is first order in $|a|/R$.

Take a transverse offset $\mathbf{b}\cdot \mathbf{R}=0$. Its distance from the source changes only at second order, but the direction of the gravitational force changes at first order:in its first-order vector expansion. Subtracting the reference acceleration givesIt compresses the transverse separation. Together with the radial result, this is point-mass tidal stretching and squeezing, with tensor eigenvalues $(2,-1,-1)GM/R^3$.

The extra radial term. A circular Kepler orbit has ${\Omega }_{\mathrm{orb}}^2=GM/R^3$. In the frame rotating with the synchronously spinning body, centrifugal acceleration increases by ${\Omega }_{\mathrm{orb}}^{2}a$ across a radial offset $a$. This adds the outward differential accelerationEquivalently, in the inertial frame the farther corotating point needs an extra inward centripetal acceleration ${\Omega }_{\mathrm{orb}}^{2}a$, which gravity alone does not supply. The total radial stretching relevant to the static corotating body is therefore $3GMa/R^3$. This is the corotating point-mass tidal field. The source's “force” expression is per unit mass.

Self-gravitational survival. Denote the asteroid radius by $s$, distinct from the orbital radius $R$. Its mass is $m_a=4\pi \rho s^3/3$ and its surface self-gravitational acceleration is $G{m}_a/s^2=4\pi G\rho s/3$. Compare this restoring acceleration with radial tidal stretching $3GMs/R^3$:The corotating tidal-disruption estimate for a uniform sphere isThis is the requested spherical local force-balance estimate, not an exact fluid Roche figure. It neglects deformation and material cohesion.

Why the asteroid radius cancels. At fixed density, self-gravitational surface acceleration scales as $m_a/s^2\propto s$. The linear tidal acceleration also scales as $s$. Their ratio is therefore independent of $s$, provided $s\ll R$ so that the tidal expansion is valid. Density, rather than asteroid size, controls this estimate.

The terrestrial value. Convert $\rho =2\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$ to $2000\mathrm{kg}{\mathrm{m}}^{-3}$ and use $M_E=6.0\times 10^{24}\mathrm{kg}$:This is distance from Earth's centre, not height above its surface.