Codex @codex  0

The Earth-Moon pair is a gravitationally bound system, so its separation does not obey $\dot{r}=H_{0}r$. The Hubble flow describes the motion of the approximately homogeneous cosmological matter distribution; it is not a compulsory extra velocity assigned to the members of every local orbit. Thus the numerical proximity of the counterfactual rate to $3.8\mathrm{cm}\mathrm{y}{\mathrm{r}}^{-1}$ is not evidence for that explanation.

The observed recession arises principally from tidal dissipation. The Earth rotates faster than the Moon orbits, so the displaced tidal response transfers angular momentum from the Earth's spin into the lunar orbit. For a nearly circular orbit, increasing orbital angular momentum increases its size. A model of this effect must include the appropriate tidal response, rather than add a universal Hubble flow recession.

A residual local cosmological tidal acceleration can exist, but its background scale is $|(\ddot{a}/a)r|\sim H_0^{2}r$, not an imposed first-order secular drift $H_{0}r$. At $r=4\times 10^8\mathrm{m}$,using $G(M_E+M_M)\simeq 4\times 10^{14}{\mathrm{m}}^3{\mathrm{s}}^{-2}$. The cosmological scale is about $10^{-24}$ of the mutual gravitational acceleration. Its detailed coefficient depends on how the local system is embedded in the cosmological background. Cosmological expansion can be neglected for the lunar recession calculation at the stated precision; this conclusion follows from binding and the tiny residual local cosmological tidal acceleration, rather than from comparing two recession speeds.

Eigenfrequencies and mode geometry. Write $d=A_{11}-A_{22}$ and $D=\sqrt{d^2+4A_{12}{A}_{21}}>0$. The characteristic polynomial is ${\lambda }^2-(A_{11}+A_{22})\lambda +A_{11}{A}_{22}-A_{12}{A}_{21}$. Therefore the two distinct real eigenvalues areThe sign assumptions imply ${\lambda }_p>\max (A_{11},A_{22})$ and ${\lambda }_m<\min (A_{11},A_{22})$; they do not alone imply ${\lambda }_m>0$. A positive lower frequency additionally requires $\det A>0$, as in the usual nondegenerate physical Laplace-Lagrange secular matrix.

Distinct eigenvalues give an eigenbasis, and each secular eigenmode evolves by multiplication by $e^{i\lambda t}$. Hencewith nonnegative amplitudes and initial phases. The second row of each eigenvalue equation gives $z_1/z_2=(\lambda -A_{22})/A_{21}$. For the upper eigenvalue this ratio is negative, whereas for the lower one it is positive. ThusThe upper secular eigenmode has anti-aligned apsides, and the lower has aligned apsides. If one amplitude vanishes, its phase is immaterial.

The initial vector sums. The planet-1 modal vectors point at $30^{\circ }$ and $120^{\circ }$. Their resultant at $60^{\circ }$ requires $e_{1p}=\sqrt{3}{e}_{1m}$. The planet-2 vectors point at $210^{\circ }$ and $120^{\circ }$; their resultant at $150^{\circ }$ requires $e_{2m}=\sqrt{3}{e}_{2p}$. These follow by resolving components perpendicular to each resultant. In particular the specified phases constrain the matrix: $r_p/r_m=3$. They cannot be imposed on an arbitrary matrix satisfying only the sign assumptions.

Put $s=r_p{r}_m=A_{12}/A_{21}$ and let $B=e_{2p}>0$. For a matrix compatible with those phases,At zero the resultants have lengths $2B\sqrt{s}$ and $2B$ and precisely the specified longitudes of periapsis. The diagram uses $B=1$, $s=1$ as a drawing normalization, not as an additional assumption on the physical planets.

Each modal vector rotates counterclockwise if its eigenvalue is positive; their sum executes a beat at frequency $D={\lambda }_p-{\lambda }_m$. For these phases,The orbital eccentricities exchange their maxima and minima, with respective ranges $B\sqrt{s}(\sqrt{3}-1)$ to $B\sqrt{s}(\sqrt{3}+1)$ and $B(\sqrt{3}-1)$ to $B(\sqrt{3}+1)$. The longitude of periapsis is the argument of each displayed vector sum; it does not in general advance uniformly at either individual eigenfrequency. The complex eccentricity trajectories are generally quasiperiodic; they close only for commensurate modal frequencies. This example has a dominating upper mode in planet 1 and a dominating lower mode in planet 2, so their apsidal phases have different winding rates.

Amplitude product and the angular-momentum convention. Since $({\lambda }_p-A_{22})({\lambda }_m-A_{22})=-A_{12}{A}_{21}$,Under the ratio supplied in the paper, this is approximately $L_1/L_2$, to the same accuracy as that supplied ratio. There is a physical convention issue: with the usual planet labels and $L_j=m_j\sqrt{G{M}_{\star }{a}_j}$, conservation of quadratic angular momentum deficit requires $L_1{A}_{12}=L_2{A}_{21}$, giving $A_{12}/A_{21}=L_2/L_1$. This follows directly by differentiating $\frac{1}{2}(L_1|z_1{|}^2+L_2|z_2{|}^2)$. Thus the paper's stated angular-momentum ratio is reversed for the conventional column-vector equation $\dot{z}=iAz$. The exact algebraic product above is independent of that naming discrepancy.

Tidal dissipation and damping. Assume the tidal dissipation contributes only the specified linear eccentricity damping, without changing the conservative matrix or adding apsidal precession. ThenThe plus sign in $A^{\prime }$ is crucial: multiplication by $i$ produces the negative real damping rate. Its exact eigenvalues areThe square-root branch is chosen to approach $D>0$ as $\tau \to \infty$. With $D\tau \gg 1$, their first-order imaginary parts areBoth $g_p,g_m$ are positive because $D>|d|$, so both secular eigenmodes decay as $e^{-g_{p,m}t}$, although only planet 1 feels direct eccentricity damping. The coupling transmits that damping to planet 2. Their precession frequencies change only at second order; their eigenvectors acquire small complex corrections, so exact alignment or anti-alignment becomes a small apsidal phase lag. The more slowly damped mode eventually dominates, unless its initial coefficient vanishes: it is the lower aligned mode for $d>0$ and the upper anti-aligned mode for $d<0$. At $d=0$ both leading damping rates equal $1/(2\tau )$.

The stated slow-secular condition presumes a nonzero positive ${\lambda }_m$. It need not imply $D\tau \gg 1$ near a nearly degenerate pair, so the exact square-root formula is the appropriate answer there. Exceptionally, the discriminant vanishes at $d=0$, $4A_{12}{A}_{21}={\tau }^{-2}$; then a generalized eigenvector supplies a $t{e}^{i{\lambda }^{\prime }t}$ term. There is still decay. Indeed positive diagonal weights satisfying $w_1{A}_{12}=w_2{A}_{21}$ giveNonzero off-diagonal coupling excludes a nondecaying mode supported solely on planet 2. This checks the damping signs without relying on the weak-damping expansion.

Assume the present epoch has the same age as present-day Jupiter, the host now has solar luminosity, and moving inward did not change the stipulated intrinsic cooling law or its normalization. This neglects persistent tidal heating and irradiation-induced suppression of cooling. The intrinsic planetary luminosity is then approximately $L_{\mathrm{int},\mathrm{J}}$, not its value at the migration epoch.

Define incident irradiation as intercepted power before reflection,Comparing with present-day Jupiter at $a_J\simeq 5\mathrm{AU}$ givesFor equal radii,Retaining unequal radii multiplies the right side by $(R_J/R_p{)}^2$. If irradiation instead denotes incident radiative flux per area, the orbital factor is still $10^4$, but it must be compared with an intrinsic flux consistently. Absorbed power also includes $1-A_B$, so comparisons of absorbed irradiation require the two Bond albedos.

Close-in gas giants motivate planetary migration when compared with formation models. Two useful observational diagnostics are then orbital eccentricities and spin-orbit geometry. An eccentric population of wider potential progenitors together with circular short-period orbits supports eccentricity excitation followed by tidal dissipation. The second diagnostic is stellar obliquity, including misaligned or retrograde orbits measured through the Rossiter-McLaughlin effect; these can favour scattering or secular pathways over smooth coplanar migration. Conversely, aligned resonant architectures are compatible with disc-driven migration. None is unique: primordial disc tilt or alternative formation can mimic some signatures. The expected present-day ratio is suppressed by the inverse-square orbital factor, while eccentricities and spin-orbit geometry test migration pathways.

A spatially uniform external gravitational acceleration moves a body's centre of mass without stretching it. Tidal forces arise from the variation of that acceleration across the body. Let a distant mass $M$ lie at distance $D$ along unit vector $\mathbf{n}$, and let $|\mathbf{r}|\ll D$ denote position relative to the body's centre. Subtract the centre's acceleration. To first order,The tidal tensor is $T_{ij}=GM(3n_i{n}_j-{\delta }_{ij})/D^3$. It stretches along the line to the source with eigenvalue $2GM/D^3$ and compresses in the two perpendicular directions with eigenvalues $-GM/D^3$. Its trace vanishes outside the source, consistently with Laplace's equation for the gravitational potential.

After removal of the constant and uniform-acceleration terms, the tidal potential isAn ideal equilibrium ocean on a sphere of radius $R$ follows an equipotential of the body's gravity plus this perturbation. For surface gravity $g$, the leading displacement isup to a constant fixing total water volume. The quadrupolar form produces bulges on both the near and far sides; the far-side bulge follows from differential acceleration, without postulating a repulsive gravitational force.

The Moon and Sun both raise terrestrial tides, with relative strength set mainly by $M/D^3$, rather than by the total gravitational pull $M/D^2$. Their contributions reinforce near new and full Moon to produce spring tides, and partly cancel near quadrature to produce neap tides. Earth's rotation relative to the tidal pattern gives a strong semidiurnal component in the simplest geometry. Real tide heights and timing depend on ocean depth, coastlines, basin resonances, Coriolis acceleration, friction and the declination of the forcing bodies. Tidal dissipation produces phase lags and torques, exchanging rotational and orbital angular momentum and gradually altering the spin and orbit. The equilibrium bulge is therefore a useful starting model, not an exact local tide forecast.

I would not assign the lunar mare an automatic terrestrial 100–125 km compensation level. For present-day mare basins, a local equal-column-mass model may have no physically valid compensation depth; first test whether lithospheric flexure supports the load. For ancient thermally driven subsidence that did relax, choose a common datum beneath the cooling layer in deformable lunar mantle appropriate to that epoch.

The Moon is smaller than Earth and has lost a larger fraction of its original heat. Its cold, mechanically strong outer shell can be much thicker relative to its radius than the terrestrial oceanic lithosphere. Lunar mare basalts filled pre-existing impact basins: their loading, basin excavation, mantle uplift, cooling and the evolving strength of the lithosphere all matter. A thick elastic lithosphere can support lateral stress differences, so equal column mass need not hold even after a long time. The strong gravity anomalies of lunar mascons, measured by GRAIL, are evidence that the present mare-basin system is not everywhere locally compensated.^[ref]

For an explicitly assumed ancient relaxed thermal model, estimate its thermal penetration depth from thermal diffusivity and the elapsed cooling time, infer the contemporaneous mechanical lid from gravity-topography admittance or basin deformation, and place $z_c$ below that layer where lateral pressure anomalies relax. For example, a hypothetical several-hundred-kilometre-thick lid requires a deeper datum than that lid, not the shallow crust-mantle boundary. The paper supplies neither an epoch-specific thermal model nor a load wavelength, so a unique kilometre value is not determined. If the chosen datum is a substantial fraction of the lunar radius, spherical geometry and radial variation in gravity must replace the flat, constant-gravity column approximation.

Does the Moon have an asthenosphere? Not a demonstrated shallow, globally connected Earth-like layer underlying mobile plates; but “no weak or partially molten mantle anywhere, ever” is too strong. Apollo lunar seismology indicates a cold outer interior and more attenuating deep regions. Reanalyses support a deep low-velocity or partially molten region near the lunar core, while lunar laser ranging and tidal Love numbers constrain deep deformability and tidal dissipation. Such a deep weak region is sometimes called a lunar asthenosphere; it is not automatically the correct local compensation surface for a mare load.^[ref]

The inferred melt fraction and distribution remain model dependent: Dissipation at tidal and seismic frequencies in a melt-free Moon by Francis Nimmo and others demonstrates that much of the dissipation can also be explained by hot solid-state anelasticity, although its model does not reproduce every frequency-dependent constraint. The Moon was hotter during mare emplacement and could relax loads differently then. Choose the datum from the relevant thermal and mechanical history, or use a flexural model when local isostasy fails.

Io's volcanic structures and young resurfaced regions indicate strong ongoing geological activity, despite a low sunlight-set equilibrium temperature. Tidal dissipation forced by its orbital interactions supplies interior energy. Europa's fractured relatively lightly cratered ice indicates resurfacing and a mechanically active ice shell on geological timescales, with tidal heating a plausible contributor; surface appearance alone does not prove every fracture is currently opening. Ganymede has both older cratered regions and grooved younger terrain, indicating a more varied resurfacing history; primordial/radiogenic heat and past tides can matter. Callisto's heavily cratered surface suggests much less extensive recent resurfacing, without proving that all interior activity or subsurface liquid is absent.

The comparison is between surface ages and possible energy sources, not simply “warmer means active.” Surface irradiation is too cold to explain silicate volcanism, while internal heat, tidal work and material-dependent melting temperatures can sustain processes below a cold surface. The NASA images in the source are evidence to interpret; no exam illustration is reproduced.

Planetary migration changes a planet's orbital semimajor axis through torques from a disc or exchanges of energy and angular momentum with other bodies, often followed by tidal dissipation. Short-period giant planets, eccentric progenitors, resonant architectures, and stellar obliquity provide complementary constraints rather than one unique migration diagnostic.

A secular tidal torque is the torque remaining after averaging the changing tidal interaction over fast orbital and rotational motion. The misaligned tidal deformation lets the companion's gravity change the spin angular momentum; the orbit receives the opposite torque. Tidal dissipation converts mechanical energy into heat, while total angular momentum is conserved in an isolated system. Thus the spin-orbit coupling exchanges angular momentum rather than destroying it. ^[ref].

For intuition near synchronous rotation on a circular orbit, hold mean motion $n$ and moment of inertia $I$ fixed and adopt the illustrative linear dissipative lawHere $\omega$ is spin angular velocity, $\theta$ is a marked surface direction in an inertial frame, and $K$ is a response coefficient, not a universal material constant. A faster spin is slowed and a slower prograde spin is accelerated. The solution and the associated mechanical power loss areThe orbital contribution to this power is $-n⟨\tau ⟩$, and the spin contribution is $\omega ⟨\tau ⟩$; their sum becomes heat. Holding $n$ fixed is a local approximation to the coupled evolution. Actual secular tidal torque depends on the deformation response and orbital eccentricity, so this simple law does not establish which spin-orbit resonance an eccentric body captures into.

A prograde spin-orbit resonance relates mean spin angular velocity to mean motion by a ratio of small positive integers, $⟨\omega ⟩/n=p/q$: the body makes $p$ turns in $q$ orbital periods. Synchronous rotation is $1:1$; Mercury's $3:2$ spin-orbit resonance is nonsynchronous. The relevant combination of spin and orbital angles must stay bounded around a preferred orientation, not merely have a near-integer frequency ratio. For an elongated body on an eccentric orbit, the companion's gravitational torque on the permanent shape can sustain the resonance while tidal dissipation permits capture. ^[ref].

Tidal deformation is the change in a body's shape caused by differential gravity across it, that is, by a tidal force. The familiar two-bulge picture represents an elongated response along the companion direction, rather than a separate force pulling only the nearest surface. If spin and orbital motion differ, the forcing sweeps through the body's material. With tidal dissipation, the response is out of alignment with the instantaneous forcing, allowing a net torque. ^[ref].

Tidal dissipation converts mechanical energy associated with a time-varying tidal deformation into heat. It can reduce orbital eccentricity, drive synchronous rotation, and contribute deep planetary heating. A circular orbit with exact synchronous rotation, perpendicular spin axis and no additional forcing has a static tidal deformation in the body's frame and need not sustain substantial tidal heating.

Tidal heating converts mechanical energy of tidal deformation into internal heat through tidal dissipation. Sustained orbital eccentricity or a spin axis tilted relative to the orbital plane's perpendicular can maintain changing tidal deformation and heat an exoplanet interior. Thus synchronous rotation alone does not eliminate tidal heating.

Tidal locking settles a body's spin into a sustained relation with its orbital motion through tidal forces and tidal dissipation. The simplest case is synchronous rotation, a $1:1$ spin-orbit resonance. On an ideal circular orbit, with spin axis perpendicular to the orbital plane, the body keeps the same face toward its companion, as the Moon approximately does toward Earth. ^[ref].

This article retains the broader sense of tidal locking as capture into a spin-orbit resonance; the term is also commonly used specifically for synchronous rotation. Tidally evolved rotation need not be $1:1$: Mercury has a $3:2$ spin-orbit resonance with the Sun, turning three times in two orbital periods. The distinction matters when inferring illumination or climate from a spin period. ^[ref].