Notes for The Moon and Mercury (Ch. 8) and Tides (Ch. 7)
7.6 The Tides
Earth has abundant surface water; about three-quarters of the surface is covered by water with an average depth of ~3.6 km. Only ~2% of Earth’s water is in lakes, rivers, clouds, and glaciers; ~98% is in the oceans.
Tides: daily fluctuations in coastal sea level, typically two low tides and two high tides per day. Typical open-ocean tide height is about 1 meter, but tides can be much larger when water is funneled into narrow openings (e.g., river mouths).
Example: Bay of Fundy tidal range can approach ~20 m (≈60 feet).
Tidal energy: daily ocean motion contains enormous energy that erodes and reshapes coastlines; in some places tides have been harnessed to generate electricity.
Gravitational Deformation
What causes tides? The Moon and the Sun gravitationally influence Earth, producing daily, monthly, and yearly tidal cycles.
Moon–Earth interaction (simplified): gravity strength depends on distance. The Moon’s gravity is stronger on the near side of Earth than at the far side by about 3% (Earth’s diameter is ~12,800 km).
This differential force creates a tidal bulge: Earth becomes slightly elongated with the long axis toward the Moon. Oceans deform most because liquids can move more easily on the surface; a bulge is also raised in solid Earth, but the solid bulge is ~100× smaller than the ocean bulge.
Daily tides occur as Earth rotates under this tidal deformation.
The Moon’s tidal effect is a differential force, diminishing rapidly with distance as the inverse cube of separation: the tidal force ∝ 1/r^3. If Earth–Moon distance doubled, tides would decrease by a factor of 8.
The Moon’s gravity causes the near-side bulge toward the Moon and a far-side bulge away from the Moon, leading to two bulges and thus two high tides per day.
The Moon and Sun both exert tidal forces on Earth; the Sun is much farther away but far more massive, producing about half the tidal effect of the Moon.
Tidal cycle interplay:
Spring tides occur when Sun and Moon gravitational forces reinforce each other (new or full Moon); tides are higher.
Neap tides occur when Sun and Moon partly cancel (first and third quarters); tides are smaller.
Effects on Earth’s rotation:
Earth’s sidereal day is 23 h 56 m, but Earth’s rotation is slowing due to tidal interactions, lengthening the day by ~1.5 ms per century.
Fossil evidence (coral growth rings) shows yearly/seasonal variations in past spin rates and day lengths.
The tidal bulge is not perfectly aligned with the Moon due to friction in Earth’s crust/ocean and interior, causing the bulge to drift slightly ahead of the Moon’s position. This drag leads to a transfer of angular momentum from Earth to the Moon, slowing Earth’s rotation and causing the Moon to recede.
The Moon is gradually moving away from Earth at ~4 cm per year.
Long-term fate: If this continues, Earth’s rotation could become synchronized with the Moon’s orbital period after many billions of years, causing the Moon to hover over the same point on Earth. At that time, Earth’s day would lengthen to about 47 current days, and the Moon’s distance would be ~550,000 km.
Concept Check
Tidal forces differ from inverse-square gravity in that tides arise from differential gravitational forces across an extended body (a gradient), not from a single central force that falls off as 1/r^2. The tidal effect scales roughly as the inverse cube of distance (1/r^3).
8 The Moon and Mercury
8.1 Orbital Properties
The Moon and Mercury are studied with detailed orbital and physical data in Moon Data (p. 202) and Mercury Data (p. 220).
The Moon’s distance from Earth:
Parallax and radar ranging give mean distance ≈
d_{ ext{Moon}} \,=\, 3.84\times 10^{5}\text{ km}Actual distance varies with the Moon’s slightly elliptical orbit.
Radar ranging yields distances with very high precision (sub-centimeter to centimeter accuracy).
Current lunar distance and orbital parameters:
Orbital semimajor axis: ~384,000 km (range ~363,000–406,000 km).
Orbital speed: ~1.02 km/s.
Sidereal orbital period: ~27.3 days.
Synodic orbital period (new moon to new moon): ~29.5 days.
Orbital inclination to the ecliptic: ~5.2°.
Greatest angular diameter seen from Earth: ~32.9′.
Moon physical properties (from NASA Moon Data table):
Mass: $7.35\times 10^{22}\ \text{kg}$.
Equatorial radius: $1738\ \text{km}$.
Mean density: $3.34\ \text{g cm}^{-3}$.
Surface gravity: $g \,=\, 1.62\ \text{m s}^{-2}$.
Escape speed: $v_{esc} \,=\, 2.38\ \text{km s}^{-1}$.
Sidereal rotation period: 27.3 days (synchronous with orbital period).
Axial tilt: ~6.7°.
Surface magnetic field: no detectable global field.
Surface temperature: ~100–700 K (extreme range; lunar day ~14 Earth days, night ~14 Earth days).
The Moon’s small size and weak gravity imply a weak or absent global magnetic field and an almost negligible atmosphere.
Mercury’s distance/orbit (
The Moon and Mercury both are inner solar-system bodies with distinct orbital characteristics. Mercury’s orbit has a semimajor axis of ~0.39 AU with an eccentricity of ~0.206. Its perihelion is ~0.31 AU and aphelion ~0.46 AU. The mean orbital speed is ~47.9 km/s; sidereal orbital period ~88.0 days; synodic period ~115.9 days; orbital inclination to the ecliptic ~7.0°; greatest angular diameter as seen from Earth ~13″.
Mass: ~$3.30\times 10^{23}\text{ kg}$; Equatorial radius: ~$2440\text{ km}$; Mean density: ~$5.43\text{ g cm}^{-3}$; Surface gravity: ~$3.70\text{ m s}^{-2}$; Escape speed: ~$4.2\text{ km s}^{-1}$; Sidereal rotation period: ~58.6 days; Axial tilt: ~0°; No natural satellites.
The Moon and Mercury are both airless bodies with extreme temperatures and heavily cratered surfaces, but Mercury shows intercrater plains and large scarps not found on the Moon.
8.2 Physical Properties
From Earth, the Moon’s angular diameter ~0.5°, enabling size estimation: Moon radius ≈ 1738 km (precise measurements give 1737–1738 km).
Mercury’s apparent radius from Earth and distance imply a radius ≈ 2440 km (≈0.38 Earth radii).
Masses and densities give clues to interiors:
Moon density ~3300 kg/m^3 vs Earth ~5500 kg/m^3; Moon is depleted in heavy elements, implying a small iron core relative to Earth.
Mercury density ~5400 kg/m^3, implying a large iron core; Mercury’s interior is highly differentiated with a substantial core relative to its size.
Gravity: Moon ~1/6 Earth; Mercury ~0.38 Earth gravity.
Atmospheres: Both Moon and Mercury lack appreciable atmospheres due to weak gravity and high daytime temperatures; trace exospheres of sodium and potassium exist.
Mercury can capture temporary hydrogen/helium from the solar wind; the Moon and Mercury have extremely tenuous exospheres.
8.3 Surface Features on the Moon and Mercury
Lunar terrain:
Maria (seas): flat, basaltic plains formed from ancient lava flows; 14 maria identified; largest mare is Mare Imbrium (~1100 km diameter).
Highlands: lighter-toned, aluminum-rich, less dense rocks; crustal material; higher elevations; older than maria.
Maria rock is basalt-like, shared characteristics with Earth’s basalt but formed in the Moon’s mantle via upwelling material.
Ages from radiometric dating: highlands ~4.0–4.4 Ga; maria ~3.2–3.9 Ga (younger than highlands).
Far side: predominantly highlands, with no large maria; suggests crustal thickness differences relative to near side.
Craters: formed by meteoritic impacts; resolution improves from ~1–2 km by Earth-based telescopes to ~500 m by spacecraft; cratering rates reveal the Moon’s bombardment history.
Lunar rock data: highlands are crust; maria rocks are mantle-derived basalt.
Surface of Mercury:
Heavily cratered like the Moon, but craters are less densely packed; extensive intercrater plains exist (not true maria).
Crater walls generally lower than Lunar craters; ejecta lands closer due to Mercury’s higher gravity.
Some older craters appear to have been flooded by volcanism, but intercrater plains differ from lunar maria in color and morphology.
Scarps (cliffs) across Mercury’s surface indicate global contraction as the interior cooled; scarps formed about 4 Ga ago.
Caloris Basin: enormous impact basin (~1400 km across) with concentric mountains around its rim; terrain on opposite side (weird terrain) likely caused by seismic waves traveling through Mercury and converging at the far side, producing disrupted terrain.
Few large lava-flow regions like the Moon’s maria; Mercury’s volcanism was earlier and less extensive in shaping surfaces.
8.4 Rotation Rates
Moon’s rotation: synchronous with its orbit around Earth (tidally locked), rotation period equals orbital period: ~27.3 days.
Result: Earth sees the same lunar face; from Moon, Earth stays nearly fixed in the sky.
Tidal torque from Earth, plus the Moon’s deformation, produced the locking; large lunar bulge indicates past stronger tides when the Earth–Moon distance was smaller (~250,000 km).
Mercury’s rotation: not synchronous with its orbit; instead, Mercury is in a 3:2 spin-orbit resonance.
Radar measurements (Arecibo, 1965) showed Mercury’s rotation was not 1:1 with its orbital period, but 3 rotations for every 2 orbits around the Sun.
Why 3:2? Mercury’s orbital speed is fastest at perihelion and slowest at aphelion due to its eccentric orbit, making a synchronized 1:1 resonance unstable; tidal forcing tends to synchronize rotation with orbital speed, but since tidal forces decrease rapidly with distance, Mercury settles into a 3:2 resonance where at perihelion it is rotating quickly enough to keep bulge aligned with instantaneous orbital velocity.
Mercury’s solar day (noon-to-noon) is two Mercury years long, meaning the Sun stays overhead for about 2 Mercury years of time in a given location, followed by roughly two Mercury years of darkness.
The Sun’s tidal effect also orients Mercury’s rotation axis perpendicular to its orbital plane.
For observers on Mercury, some longitudes become the hottest (hot longitudes) when the Sun is overhead at perihelion; poles can have temperatures as low as ~125 K, potentially hosting surface ice in permanently shadowed regions.
8.5 Lunar Cratering and Surface Composition
The Moon lacks atmosphere and water erosion; meteoroids are the primary agents shaping the surface through cratering.
Meteoroid impacts occur at speeds of several km/s; a small object carries enormous energy (e.g., a 1-kg object at 10 km/s releases energy comparable to 10 kg of TNT).
Crater formation:
Final crater diameter is typically ~10× the meteoroid diameter; depth is ~2× meteoroid diameter.
Impact creates shock waves and melts rock; ejecta blankets surround craters; larger ejecta can create secondary craters.
Micrometeoroids also erode the surface, leaving glassy beads in lunar regolith.
Cratering rate and erosion:
Current rate: ~1 new 10-km crater per 10 million years; a meter-sized crater about once per month; centimeter craters every few minutes.
Erosion rate is slow: ~5 m per billion years; far slower than Earth’s erosion.
Lunar regolith (lunar dust): about ~20 m thick on average; finer near maria, thicker in highlands; regolith forms from continuous micrometeoroid bombardment.
Lunar cratering history:
4.0–3.9 Ga era of heavy bombardment; highlands formed before/through this period.
Late heavy bombardment around 4.1–3.9 Ga flooded many basins with lava, forming the maria.
Orientale Basin (~3.9 Ga) is a younger, unflooded basin; other basins on the far side are also unflooded.
Lunar ice:
There is evidence for water ice at the lunar poles; radar data (Clementine) suggested ice deposits in permanently shadowed craters.
Lunar Prospector detected hydrogen showing water-ice signatures; later analyses proposed that much of the ice might be in concentrated “lakes” slightly below the surface, at depths of a few tens of centimeters to a few meters, with large amounts possibly present.
Ice deposits could be crucial for future lunar exploration and colonization due to water being a valuable resource.
Lunar volcanism and interior:
Early Moon had significant volcanism that produced maria; later volcanic activity subsided by ~3 Ga.
Apollo samples and Lunar Prospector data helped constrain the Moon’s internal structure and differentiation history.
Lunar history highlights (from Section 8.9):
Moon formed ~4.6 Ga ago; oldest highland rocks ~4.4 Ga.
Early bombardment heated and melted surface layers; differentiation likely occurred early; possible small iron core forming around this time.
Heaviest bombardment ended ~3.9 Ga; maria formed ~3.2–3.9 Ga; youngest maria ~3.2 Ga.
Cratering rates have declined since then, but micrometeorite impacts continue.
Lunar interior model (8.7):
Core may be small and possibly partially molten; inner mantle ~400 km thick; outer mantle ~900–950 km; crust ~60–150 km thick (near side thinner than far side).
Lunar crust is thinner on the near side (Earth-facing) and thicker on the far side, due to gravitational effects during differentiation.
Discovery boxes (Discovery 8-1 and 8-2) summarize unmanned lunar exploration:
Clementine (1994): a small (≈150 kg) mission by the U.S. Defense Department to test lightweight spacecraft technologies; produced a digital global map of the Moon and demonstrated lidar.
Lunar Prospector (1998): 295 kg orbiter focusing on mapping elemental abundances, mapping magnetic and gravity fields, and searching for water ice via neutron spectrometry.
Both missions demonstrated the value of low-cost, rapid-turnaround space exploration and provided new insights into the Moon’s interior, polar ice, and surface history.
8.6 The Surface of Mercury
Mercury’s surface is similar to the Moon in its cratering-dominated morphology but with key differences:
Craters are less densely packed; more intercrater plains exist.
Crater walls are generally not as high; ejecta lands closer to the impact site due to Mercury’s stronger surface gravity than the Moon’s.
Older craters may have been partially filled by volcanism; intercrater plains are lighter in color and not as lava-rich as lunar maria.
Scarps on Mercury: large cliffs likely formed when Mercury’s interior cooled and contracted, producing long, high scarps.
Caloris Basin: a massive impact basin (~1400 km across) with concentric mountains; reflections of seismic waves from Caloris caused odd terrain on the opposite side of the planet (weird terrain).
Mercury’s lack of atmosphere and extreme temperature variations:
Daytime temperatures can reach ~700 K near the equator; nighttime temperatures drop to ~100 K.
Polar regions can harbor water ice in permanently shadowed regions due to extremely low temperatures.
Knowledge gained from spacecraft:
Mariner 10 (1970s) provided the first high-resolution images; mapped less than half of Mercury’s surface due to its 3:2 resonance and flyby geometry.
The planet’s surface features and Caloris Basin reveal a history of early melting and extensive bombardment.
8.7 Interiors
The Moon’s interior (from Section 8.7):
Density of Moon’s crust and mantle: crust ~2900 kg/m^3; maria mantle material ~3300 kg/m^3.
The Moon likely has a small iron core, possibly ~300 km in radius; core is probably partially molten, but temperature estimates vary.
Seismic data from Apollo missions suggest a partially molten inner core and a solid outer core or a fully molten core with a solid inner region.
The Moon’s lack of a strong global magnetic field is consistent with a slow rotation rate and a possibly partially molten core; a fully liquid iron core is not required to produce magnetism.
The Moon’s crust is thicker on the far side, possibly due to Earth’s gravity pulling mantle material toward Earth, making the near-side crust thinner.
Mercury’s interior (from Section 8.7):
Mercury is highly differentiated with a large, dense iron core; the core may account for ~60% of Mercury’s mass and ~40% of its volume, with a radius perhaps ~1800 km.
Above the core lies a mantle, approximately 500–600 km thick, and a relatively thin crust.
The presence of a magnetic field (though weak) implies a partially molten core and a dynamo may be possible or may be a remnant of an extinct dynamo.
Mercury’s large core relative to size and high mean density distinguish it from the Moon and from Earth’s composition.
8.8 The Origin of the Moon
The origin of the Moon has been debated; several theories existed:
Sister/coformation theory: Moon formed alongside Earth from the same material, like a double planet.
Capture theory: Moon formed elsewhere and was captured by Earth’s gravity.
Daughter/fission theory: Moon torn away from a rapidly spinning young Earth (e.g., Pacific basin origin).
All three theories face problems when accounting for the Moon’s specific composition and isotopic similarities/differences with Earth.
Current favored hypothesis: the giant-impact (hybrid capture/daughter) scenario:
A Mars-sized body collided with a young Earth; debris reassembled to form the Moon.
The impact explains similarities between the Moon’s mantle and Earth’s mantle, while the lack of a dense central core in the Moon is explained by the collision ejecting the metallic core into Earth.
Computer simulations (Figure 8.28) show the potential sequence of a large impact producing a Moon-sized body that coalesces from debris.
The condensation theory provides a natural context for such collisions in the early solar system.
Important caveats:
Some aspects remain debated, such as the degree to which the Moon melted during formation and whether current models fully match observed lunar composition.
The Moon provides a unique case; exomoons may have different formation histories.
Concept Check: The Giant-Impact theory accounts for the Moon’s low heavy-element content relative to Earth and its similarity in mantle composition to Earth, due to the dominance of rocky material reassembled from debris rather than a pristine copy of Earth’s entire bulk.
8.9 Evolutionary History of the Moon and Mercury
The Moon’s evolutionary history:
Formation: ~4.6 Ga ago.
Oldest lunar rocks: ~4.4 Ga, indicating an early solid crust and differentiation.
Early bombardment (first ~0.5 Gyr): frequent enough to heat and melt surface layers to depths of ~400 km; likely formed a partially molten outer region and a small iron core.
End of heavy bombardment: ~3.9 Ga, followed by solidification of a crust and the formation of basins.
3.9–3.2 Ga: major volcanic activity filled basins with basaltic lava, creating maria; young maria age ~3.2 Ga indicates volcanism subsided by then.
Post-volcanic era: Moon became geologically very quiet; little to no tectonic or volcanic activity since ~3 Ga.
Mercury’s evolutionary history:
Formation ~4.6 Ga ago; rocks show depletion of lighter elements due to proximity to the Sun and high-temperature processing in the early solar system.
Early melting and differentiation occurred; large iron core formed and cooled slowly, allowing contraction and shaping of scarps.
Crustal contraction led to scarps and compression features; volcanism occurred early but less extensive than the Moon’s maria due to Mercury’s smaller mantle-crust interactions.
Mercury’s volcanic period likely ended earlier than the Moon’s; overall tectonic and volcanic activity has been low for much of its history.
Overall comparison:
Both Moon and Mercury are airless, showing extreme day–night temperatures due to lack of atmosphere and poor heat conduction.
Both show crater-dominated surfaces with extensive meteoritic bombardment in early history.
The Moon’s interior shows a smaller iron core relative to Earth; Mercury has a large iron core relative to its size, implying significant differentiation and a unique thermal history.
Summary of the evolutionary trajectories:
Moon: rapid early heating and differentiation, extensive volcanism early on, then long-term dormancy; Maria formed from mantle-derived basalt, highlands from crustal material; polar ice remains a possibility.
Mercury: early differentiation with a large iron core, thermal contraction creating scarps, less volcanic resurfacing than the Moon, possible enduring weak magnetism possibly from a partially molten or extinct dynamo; polar regions may harbor water ice in permanently shadowed craters.
Discovery and Context
Discovery 8-1: Lunar Exploration – The Space Age began with Sputnik 1 (1957); Luna missions (Luna 1–3, 1959) provided early data, including first images of the far side (Luna 3). U.S. Ranger, Lunar Orbiter, and Surveyor missions advanced mapping, imaging, and surface analysis ahead of Apollo. Apollo program (1961–1972) achieved crewed Moon landings, sample return (~382 kg), ALSEP experiments, and long-term science goals. The era highlighted the importance of robotic precursors and later demonstrated the value of sample return for understanding planetary formation and history.
Discovery 8-2: The Moon on a Shoestring – Clementine (1994) and Lunar Prospector (1998) demonstrated high scientific return for relatively low budgets (~$70 million each). Clementine produced a global Moon map across visible, UV, and IR with lidar; Lunar Prospector mapped elemental abundances, magnetic and gravity fields, and searched for water ice. These missions illustrated how small, rapid missions can deliver high-impact science and influence future planetary exploration strategies.
Connections to Foundational Concepts and Real-World Relevance
Tidal physics connect to angular momentum exchange and orbital evolution, illustrating how long-term gravitational interactions shape planetary systems and even planetary rotation rates.
Lunar and planetary interiors illustrate differentiation, heat flow, and core formation – core-mantle dynamics underpin magnetic fields, tectonism, and planetary evolution.
The Moon’s cratering history provides a robust chronometer for solar-system events and helps calibrate the timing of major bombardment episodes, informing models of early planetary formation for the entire solar system.
The Moon’s polar ice has profound implications for future human exploration and in-situ resource utilization (ISRU); the ice could be a critical resource for life support and fuel if accessible.
Mercury’s unusual 3:2 spin-orbit resonance and polar ice considerations reveal the complexity of tidal interactions under high orbital eccentricities and their multifaceted effects on surface temperature regimes and potential volatiles.
The giant-impact theory for the Moon’s origin demonstrates how cosmochemical data, dynamical simulations, and isotopic analyses combine to illuminate planet formation processes, while highlighting remaining uncertainties and the role of observational constraints in refining theories.
Key Formulas and Numerical References
Tidal force scaling with distance (concept): tidal force ∝ 1/r^3 (inverse-cube law for differential gravity).
Moon–Earth bulge geometry: tidal bulges on near and far sides caused by differential lunar gravity; Earth’s rotation carries the bulge beneath Earth’s surface, generating torque that slows Earth and recedes the Moon.
Differential gravitational effect across Earth: strength is greatest on the near side, weakest at the far side; bulges align with the Moon–Earth line.
Critical distance/ratio example: If the Earth–Moon distance doubles, the tidal effect reduces by a factor of 8.
Escape speed (Earth units):
v_{esc} ext{(km s}^{-1}) = 11.2 \, \sqrt{\frac{M}{R}}
where M is the planet’s mass in Earth masses and R is its radius in Earth radii.Average molecular speed in a gas (Earth units):
\bar{v} \,=\, 0.157 \, \sqrt{\frac{T}{M}}
where T is the temperature in Kelvin and M is the molecular mass in hydrogen atom masses.Spin-orbit resonance example for Mercury: 3:2 (Mercury completes 3 rotations for every 2 orbits around the Sun).
Mercury’s orbital and rotation parameters (selected):
Orbital semimajor axis: ~0.39 AU
Eccentricity: ~0.206
Perihelion: ~0.31 AU; Aphelion: ~0.46 AU
Sidereal rotation period: ~58.6 days
Orbital period: ~88.0 days
Axial tilt: ~0°