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View a circular orbit along its perpendicular spin axis. Let $f$ be the orbital angle measured from a fixed direction, choosing an arbitrary zero for the circular limit of true anomaly, and let $\theta$ be the direction of an outward arrow painted on the body, measured from the same fixed direction. The direction from the body to its companion is $f+\pi$. Thus an inward-facing arrow satisfiesMore generally synchronous rotation keeps $\theta -f$ constant; its particular value selects which surface direction faces the companion. Over one orbital period $T$, both $f$ and the unwrapped $\theta$ increase by $2\pi$. This is the short geometric explanation for one turn per orbit. For a nonspinning body, $\dot{\theta }=0$ instead, and the companion direction sweeps through all surface longitudes. On an eccentric orbit, mean $1:1$ spin does not force $\theta -f$ to stay exactly constant: true anomaly advances nonuniformly, producing the longitudinal component of libration.

Facing a companion and facing the source of light are different relations. The Moon keeps its near side of the Moon approximately toward Earth, while the direction to the distant Sun is nearly fixed during the idealized orbit shown below. A fixed surface patch therefore turns toward, across and away from sunlight. The far side of the Moon has day and night too; its name does not mean permanent darkness. ^[ref].

If the Sun direction has fixed angle $\alpha$ and the marked surface patch has outward normal vector at angle $\theta$, its direct illumination is proportional to $\max (0,\cos (\theta -\alpha ))$. With synchronous rotation, $\theta =f+\pi$, so the patch goes through day and night even though it always faces Earth. In reality the Earth-Moon system advances around the Sun: the lunar spin and orbital cycle relative to distant stars is about $27.3$ Earth days, while the lunar phase cycle has a synodic period of about $29.5$ Earth days. These numbers describe the actual system, not the fixed-sunlight cartoon. ^[ref].

For a synchronously rotating exoplanet whose orbital companion is itself the illuminating star, the direction to the star turns with the orbit. On an ideal circular orbit, with perpendicular spin axis and no additional oscillations, this direction is fixed relative to the surface, giving persistent day and night hemispheres. Day-night heat redistribution then helps determine the nightside temperature; permanent lack of direct starlight does not imply zero temperature. Orbital eccentricity and departures from this ideal spin geometry modify that simple picture.

Libration is the apparent rocking of a body as seen by an observer. For the Moon, longitudinal libration arises because orbital eccentricity makes its orbital angular speed vary while its spin is much more uniform: the direction to Earth advances ahead of, then falls behind, the mean facing direction. The Moon's tilted geometry also causes latitudinal libration; small actual rotational oscillations provide further corrections. Thus lunar synchronous rotation holds in the mean, rather than requiring exact instantaneous alignment. ^[ref]. ^[ref].

A lunar phase is the appearance of the sunlit portion of the Moon as seen from Earth. Outside a lunar eclipse, the distant Sun illuminates approximately one hemisphere of the Moon, but an observer sees a changing overlap of that hemisphere with the hemisphere facing Earth. The lunar phase therefore changes even though synchronous rotation keeps approximately the same near side of the Moon facing Earth. ^[ref].

The cycle runs through new Moon, waxing crescent, first quarter, waxing gibbous, full Moon, waning gibbous, last quarter and waning crescent. “Waxing” means that the lunar illuminated fraction is increasing; “waning” means it is decreasing. A crescent has less than half its apparent disk sunlit, and a gibbous lunar phase has more than half. First and last quarter each show half a disk: “quarter” refers to progress around the cycle, rather than a quarter of the disk being lit. ^[ref].

An instructive circular orbit model puts Earth at the origin, the Sun far away along $+x$, and the Moon at $R(\cos \theta ,\sin \theta )$. The sunward position $x>0$, $\theta =0$, gives new Moon; the opposite position $x<0$, $\theta =\pi$, gives full Moon. Counterclockwise motion gives first quarter at $\theta =\pi /2$ and last quarter at $\theta =3\pi /2$. The Sun's direction is fixed during this snapshot of the cycle. Every small globe in the orbital view is illuminated on its $+x$ half; the adjacent apparent disks are separate views from Earth, rather than globes viewed from above.

The disk views choose north up, with the upward direction perpendicular to the orbital plane. In this convention waxing lunar phases are bright on the right and waning lunar phases on the left. Local horizon orientation and viewing from different locations on Earth can rotate the apparent disk; its lit edge always points toward the projected direction of the Sun. The lunar illuminated fraction is unaffected by this rotation.

The useful angle is the phase angle $\alpha =\angle SME$, with $S$ the Sun, $M$ the Moon and $E$ the observer at Earth's center: its vertex is at the Moon. Full Moon has $\alpha =0$, quarter has $\alpha =\pi /2$, and new Moon has $\alpha =\pi$ in the ideal model. By contrast, the elongation $ϵ=\angle SEM$ has its vertex at Earth. For a distant Sun, its rays are parallel, soHere $k$ is the lunar illuminated fraction: a fraction of projected disk area, not a fraction of the Moon's total surface area or its received radiant flux relative to full Moon. The phase angle and unsigned elongation each lie between $0$ and $\pi$, so either alone gives the illuminated amount without distinguishing waxing from waning. The orbital direction resolves that ambiguity.

The lunar illuminated fraction has an elegant geometric derivation. Model the Moon as a sphere of unit radius and view it along the $z$ direction. Take the Sun's unit direction to be $(\sin \alpha ,0,\cos \alpha )$. A visible surface point above apparent disk coordinates $(x,y)$ has $z=\sqrt{1-x^2-y^2}$. Direct illumination requires the outward normal's dot product with the sunward direction to be positive. The day-night terminator therefore obeys $x\sin \alpha +z\cos \alpha =0$. At a fixed $y$, set $q=\sqrt{1-y^2}$; the visible day-night terminator is $x_t=-q\cos \alpha$, and the illuminated strip extends from $x_t$ to the right edge $q$. Integrating its width givesThe integral is the area of half a unit circle. Thus a three-dimensional visibility problem reduces to a circle's area and a cosine. The derivation assumes parallel sunlight, a distant observer and a sphere, and ignores a lunar eclipse. Real observed brightness additionally depends on surface scattering of light and observing geometry, so $k=1/2$ does not promise half the full-Moon radiant flux.

For a worked example, elongation $ϵ=60^{\circ }$ gives phase angle $\alpha =120^{\circ }$ and $k=(1+\cos 120^{\circ })/2=1/4$: a crescent with $25\%$ of its apparent disk sunlit. At elongation $120^{\circ }$, the phase angle is $60^{\circ }$ and $k=3/4$: a gibbous lunar phase. At $90^{\circ }$, $k=1/2$. In each case the same $k$ occurs once while waxing and once while waning in the ideal cycle.

Ordinary lunar phases are not caused by Earth's shadow. A lunar eclipse occurs only when the Moon actually enters that shadow near full Moon. The orbital inclination of the Moon is about $5^{\circ }$ relative to Earth's orbital plane around the Sun; most full Moons pass above or below the shadow. This is why full Moon does not imply a lunar eclipse every month. ^[ref].

The phase cycle is a synodic month, about $29.5$ days, whereas the Moon's orbital period relative to distant stars is a sidereal month, about $27.3$ days. During one sidereal month, Earth advances around the Sun, so the Moon must move farther to repeat the sunward alignment. For coplanar circular orbits traversed in the same direction, subtract their mean motions: $n_{\mathrm{syn}}=n_{\mathrm{sid}}-n_{\mathrm{year}}$, with $n=2\pi /P$. HenceUsing $P_{\mathrm{sid}}=27.321661\mathrm{days}$ and $P_{\mathrm{year}}=365.256\mathrm{days}$ gives $P_{\mathrm{syn}}=29.530591\mathrm{days}$, consistent with the mean synodic month of $29.53059$ days. After one sidereal month, the sunward direction has advanced about $26.93^{\circ }$; catching up takes about $2.21$ more days. These are mean-period estimates: orbital eccentricity and changing geometry make individual synodic months vary. ^[ref]. ^[ref].

The lunar phase is also a practical clock for observing. Full Moon is opposite the Sun and rises roughly at sunset; first quarter is about a right angle from the Sun and is convenient in the evening, while last quarter is convenient before dawn. These are rough guides, not exact rise-time predictions for every place and season. ^[ref]. Historically, repeated lunar phases provided the month in lunar calendars, while half-lit disks gave Aristarchus of Samos a way to infer astronomical distances. ^[ref].

The Moon is Earth's natural satellite. Its synchronous rotation keeps approximately the same near side of the Moon facing Earth, while both the near side of the Moon and the far side of the Moon experience sunlight and darkness. See illumination of a synchronously rotating moon for the geometry. ^[ref].

A natural satellite is a naturally occurring body orbiting a planet or another nonstellar body. The Moon is Earth's natural satellite; its synchronous rotation provides a familiar example of tidal locking.

The near side of the Moon is the hemisphere directed approximately toward Earth. Libration changes the exact visible boundary, so synchronous rotation does not mean that an observer always sees precisely the same hemisphere.

A close-in giant is often in synchronous rotation after tidal locking, giving persistent dayside heating and nightside cooling. Its contrast is controlled by the competition between radiative relaxation time in a planetary atmosphere, wind transport characterized by the atmospheric advection time, wave adjustment, and drag. When heat transport is fast compared with radiation, day-night heat redistribution lowers the contrast; when radiation is fast, each hemisphere stays closer to its local radiative balance.

For a rough atmospheric column estimate,At comparable pressure, higher planetary equilibrium temperature sharply shortens radiative relaxation, tending to increase the day-night contrast. Wind speeds, rotation, and magnetic drag can modify this trend; dissociation and recombination can carry additional heat in very hot atmospheres.

At higher altitude, lower pressure generally means shorter radiative relaxation and a larger contrast. Infrared bands with larger opacity probe these higher layers, while lower-opacity windows sample deeper layers with longer cooling times and more effective redistribution. A wavelength-dependent exoplanet thermal phase curve can therefore reveal how the contrast and hot-region displacement change with pressure.

The lunar crustal dichotomy combines several observables: the far side of the Moon has generally thicker, higher, feldspathic lunar highlands, whereas most visible lunar maria and the Procellarum KREEP Terrane occupy the near side of the Moon. These need not all have been produced by a single event. The old anorthosite crust formed mainly before the younger basaltic mare infill. Gravity-based crustal models combined with sample and seismic constraints demonstrate that crustal thickness and porosity must be considered alongside surface color.^[ref]

An early thermal and chemical asymmetry is one physically motivated option. In a lunar magma ocean, crystallizing, relatively buoyant plagioclase can accumulate to form feldspathic crust. On a young Moon that rapidly attained synchronous rotation, radiation from a still-hot nearby Earth would preferentially warm the nearside. The primordial Earthshine hypothesis proposes that this imposed a thermal gradient in the atmosphere and accreting material: crust-forming refractory constituents condensed preferentially toward the cooler farside, favoring thicker farside crust. This is more specific than simply saying that the farside cooled faster; it requires a persistent hemispheric chemical contrast.^[ref]

Subsequent concentration of incompatible elements in KREEP-rich late liquids can help preserve a thermal asymmetry: potassium, uranium and thorium provide radiogenic heating, supporting later melting in the nearside source region. Lower surface elevation and a thinner crust also facilitate some magma transport and pooling, although crustal thickness alone cannot specify eruption rates. The origin of the crustal contrast and the later production and emplacement of mare basalt are separate parts of the explanation.

The thermal option is consistent with a thick feldspathic farside crust, a hemispheric chemical contrast and the existence of an early hot Earth. It is not uniquely demonstrated by those observations. Vigorous magma-ocean mixing could erase the proposed gradient; the required irradiation must persist during the relevant condensation and crust-building interval, not merely occur at some early time. A failure to find the predicted bulk compositional contrast in demonstrably ancient deep crust, or a chronology placing the asymmetry after complete crystallization, would challenge this particular mechanism. The existence of farside mare deposits and large thin-crust basins also warns against treating thickness as an on/off volcanic switch.

Accretion of a smaller companion is a contrasting option. A low-speed collision of a smaller co-orbiting moon with the lunar farside could add a broad layer of relatively cold silicate material rather than excavate an ordinary deep crater. Numerical impact models demonstrate that such a depositional event can build thick highlands and displace underlying residual liquid toward the opposite hemisphere.^[ref] Broad crustal excess and asymmetric evolved material are compatible with that mechanism, but are not unique evidence for it. The necessary companion, collision timing and impact geometry must be dynamically feasible. A predicted buried depositional interface, age discontinuity or lateral fabric should survive somewhere; their systematic absence in well-resolved, unmodified ancient crust would count against the model. Closely related bodies could have similar isotopes, so failure to find a large isotopic difference alone would not disprove accretion.

Both scenarios allow later basin impacts, partial melting of the lunar mantle and mare emplacement to modify the original contrast. A useful explanation must reproduce crustal mass, chemistry, chronology and volcanic distribution together, and state which observations would distinguish its initial cause.

A proposed explanation of the lunar farside highlands in which irradiation from a nearby hot early Earth imposes a thermal gradient on a synchronously rotating, accreting Moon. Preferential condensation and deposition of refractory crust-forming material toward the cooler farside seeds a lasting chemical and crustal-thickness asymmetry. It requires early synchronous rotation, appropriate transport times and incomplete homogenization by subsequent mixing. It is a testable formation hypothesis, distinct from ordinary present-day illumination of the lunar night side.^[ref]

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 sidereal month is the Moon's orbital period relative to distant stars, about $27.32166$ days. It is shorter than the synodic month because the sunward direction changes as Earth travels around the Sun. The lunar spin period associated with synchronous rotation matches the sidereal month in the mean. ^[ref].

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].

Synchronous rotation means one prograde turn about the body's own axis per orbital period, measured against fixed directions in an inertial frame. This $1:1$ spin-orbit resonance approximately describes the Moon's synchronous rotation toward Earth. ^[ref].

The striking geometric idea is that one turn in space can keep one face toward the companion: a fixed orientation in the frame turning with the orbit is a changing orientation in an inertial frame. This distinction helps us predict illumination: a synchronously rotating exoplanet orbiting its illuminating star can have persistent day and night hemispheres, whereas the Moon, synchronized to Earth, receives changing illumination from a different body, the Sun.

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].