How Scientists Weigh the Moon

M_earth + M_moon = 4 * pi^2 * a^3 / (G * T^2) for semi-major axis a about 384,400 km and sidereal period T about 27.321661 days. Without G, Kepler’s third law gives you the sum of masses times G. With a barycenter measurement you get the ratio. Only with G can you turn the dynamical quantity into an absolute mass in kilograms.

For orbital dynamics the relevant quantity is GM rather than mass in kilograms. Spacecraft, propellant and payload masses are still expressed in kg, but orbit propagation uses gravitational parameters. Saying we never use kilograms is too absolute – the correct statement is that the dynamical quantity governing the orbit is GM.

The Moon does not orbit Earth’s geocenter. Earth and Moon orbit their common barycenter about 4,670 km from Earth’s center per JPL lunar dynamical models, with an Earth/Moon mass ratio of about 81.30057. Using mean Earth radius 6,371 km, that places the barycenter about 1,700 km below Earth’s surface. Earlier rounding of about 4,671 km is consistent within a few kilometres.

Earth’s monthly wobble about that barycenter provides the geometric basis for the ratio, but modern determinations combine astrometry, lunar laser ranging and dynamical modeling rather than obtaining the ratio directly from a single wobble measurement. The ratio itself is independent of G.

Historical context: Henry Cavendish in 1798 measured the mean density of Earth with a torsion balance, from which the gravitational constant G can be inferred. The experiment is now commonly described as a measurement of G, although Cavendish’s original objective was to determine Earth’s density.

Mass vs weight: Mass is invariant. Weight is m * g_local. Lunar surface gravity is about 1.62 m/s2 versus about 9.80 m/s2 on Earth. The hypothetical Earth-surface weight of a Moon with mass 7.346e22 kg would be about 7.20e23 N. That is a hypothetical number with no dynamical use.

Tides: Lunar tidal forcing at Earth is about 2.2 times solar tidal forcing. This provides an independent dynamical constraint when the distance dependence and tidal potential are included in the model, rather than acting as a simple separate mass measurement.

Spacecraft Doppler and related tracking observations constrain GM directly. One widely cited published lunar GM is 4902.800066 km3/s2 plus/minus 0.0001 km3/s2 at 1-sigma in the cited NASA technical document. That value is legitimate and useful as a published geophysical working value. Current JPL DE440 satellite parameter tables list lunar GM as 4902.800 plus/minus 0.001 km3/s2 tied to DE440.

So 4902.800066 plus/minus 0.0001 should be presented as a published working value rather than unqualifiedly calling it the current formal JPL uncertainty. The 2.0e-8 relative uncertainty corresponds to plus/minus 0.0001 / 4902.8. The DE440 table value corresponds to about 2.0e-7 relative.

CODATA 2022 G is 6.67430(15) x 10^-11 m3/kg/s2, with relative standard uncertainty about 2.2e-5 = 0.0022 percent. Therefore the conversion from GM to kilograms is overwhelmingly limited by G if those GM uncertainties are adopted. With plus/minus 0.0001 for GM, the ratio of relative uncertainties is about 1,100. With plus/minus 0.001, the ratio is about 110. In either case the kilogram conversion has an approximately 0.0022 percent uncertainty contribution from G.

The Lunar Orbiter program, five missions from 1966 to 1967, provided tracking that revealed systematic gravity anomalies over mare basins. The important part is correct: Lunar Orbiter tracking revealed the mascons, and Paul Muller and William Sjogren identified them in 1968. Describing all five simply as low-inclination orbiters is less precise, since the missions had different inclinations and objectives. NASA and JPL historical material supports the discovery from Lunar Orbiter tracking.

NASA Lunar Prospector documentation explicitly says its Doppler Gravity Experiment used S-band radio tracking to determine the spacecraft orbit and lunar gravity field. NASA PDS documentation also notes the limitation caused by the absence of direct farside tracking: spacecraft cannot be directly tracked from Earth over a large part of the lunar farside. Consequently, the farside gravity field was much less directly constrained before missions capable of relaying measurements around the Moon.

Doppler data quality varied. PDS documentation reports values ranging from roughly 0.3 mm/s to several mm/s depending on the data, with Clementine around 0.25 mm/s. Quoting about 0.5 mm/s is therefore reasonable only as an order-of-magnitude description rather than as a universal instrument specification.

JAXA SELENE achieved the world’s first direct observation of the lunar farside gravity field using four-way Doppler. When Kaguya was behind the Moon, subsatellite Rstar (Okina) relayed its signal to Earth, and Vstar (Ouna) provided VRAD measurements. JAXA explicitly describes the November 2007 experiment as the world’s first direct observation of the lunar farside gravity field.

That enabled a much more complete global modeled gravity field. It is better not to describe this as closing a literal 100 percent gravity field: SELENE provided direct farside observations, but it did not independently measure every point of a continuous gravity field. The resulting global field remained a model constrained by observations from multiple sources, now including direct farside data.

DSN Doppler measures carrier frequency shift. Precision is often discussed in terms of frequency or phase measurements, but conversion to line-of-sight velocity depends on carrier frequency and Doppler convention. You cannot give a single velocity equivalence for both S-band and Ka-band this way.

In practice, S-band lunar tracking achieved velocity precision at roughly the 0.1 to 1 mm/s level in the relevant historical applications. GRAIL science is properly described as inter-spacecraft Ka-band range-rate measurement, with published primary-mission data weighting around 0.03 to 0.1 micron per second. The formal observable is range-rate, not a direct sub-micron range measurement.

The GRAIL sensitivity figure is therefore best described in terms of range-rate. Published values around 0.03 micron per second correspond to extremely small changes in the separation rate of the two spacecraft. Describing the system as having roughly 0.1 micron “range sensitivity” can be useful as an order-of-magnitude shorthand, but it should not be presented as the instrument’s direct range accuracy.

Ebb and Flow flew in near-polar, near-circular orbits. Mean altitude during primary science was about 55 km, while periapsis altitude ranged about 16 to 51 km per NASA documentation. The 16 to 51 km figure refers to periapsis altitude, not to variation in the mean orbital altitude.

Separation during primary science varied from about 85 km to 225 km and then to about 65 km per NASA PDS documentation. The earlier use of 100 km was a rounded approximation; about 85 km is preferable for a technical description.

Primary gravity field GL0420A degree and order 420 is described by NASA’s public map material as resolving surface blocks about 20 km across, while technical literature describing GL0420A gives a characteristic half-wavelength of about 13 km for degree 420. These are different ways of describing spatial resolution, so they should not be treated as contradictory measurements.

Public language described the primary GRAIL gravity map as three to five times improved over previous data for particular metrics used in those releases. Technical literature describes much larger spectral improvements, reaching roughly four orders of magnitude in some spectral ranges. Thus “three to five times” should be tied to the particular public-release metric rather than presented as the overall improvement in lunar gravity knowledge.

Later JPL field GL0660B achieved regional surface resolution between about 14.6 km and 9.3 km per published JPL work. GL0660B is a major 660×660 primary-mission field, not the final word on lunar gravity modeling. Subsequent GRAIL-derived models reach higher degree and order, including the GRGM1200A/1200B family of products and later localized solutions available through NASA’s Planetary Geodesy Data Archive.

Mascons are positive free-air gravity anomalies associated with large impact basins. They are not simply mare basalt filling a hole. NASA’s GRAIL interpretation describes large impacts occurring when the Moon’s interior was much hotter, with the interaction of relatively light crust, dense mantle and impact shock producing distinctive density structures. Older work also identified contributions from mantle uplift and high-density mare basalt infill. The relative contribution varies by basin.

Free-air anomalies associated with mascons can reach hundreds of milligals. NASA GRAIL maps show roughly plus/minus 400 mGal in some presentations, while the GL0660B paper gives surface radial acceleration extrema of about +1268 and -645 mGal for a specified truncation. The exact extrema therefore depend on the model and representation being quoted.

A safer comparison with Earth is that lunar gravity variations can be substantially larger in the relevant context, rather than claiming an order-of-magnitude larger fractional variation without specifying altitude, reference field, wavelength and whether free-air or Bouguer anomalies are being compared.

Average global lunar crustal thickness from GRAIL-based work is commonly given as between about 34 and 43 km depending on the seismic constraint and model assumptions. The nearside-farside pattern is important, but 34 km and 43 km should not be treated as fixed thicknesses for the two hemispheres. More recent work continues to emphasize uncertainty associated with limited seismic constraints and assumptions about crustal composition and density.

Low lunar orbits around 50 km can require frequent station-keeping, and depending on inclination, geometry and initial conditions, mascon-driven perturbations can eventually drive a spacecraft into the surface. Lifetime is not universally measured in days.

A useful real-world example is NASA’s Lunar Reconnaissance Orbiter, which operated in an approximately 50 km mapping orbit and required regular station-keeping. NASA LROC material gives an example in which an unattended 50 km orbit would eventually impact the Moon after about 70 days. That is an example of a particular orbital configuration, not a universal lifetime for every 50 km lunar orbit.

GRAIL dramatically improved lunar gravity knowledge and therefore orbit prediction and navigation precision. It did not make low-altitude or frozen orbits possible for the first time. Low-altitude lunar orbits existed before GRAIL, and frozen-orbit concepts had also been studied before GRAIL, with LRO using a quasi-frozen orbital strategy.

The more accurate statement is that GRAIL made low-altitude orbit design, prediction, navigation and station-keeping substantially more precise by providing a much better lunar gravity model. It remains incorrect to claim that spacecraft would simply crash without GRAIL. Spacecraft successfully orbited the Moon before GRAIL, including Lunar Orbiter, Apollo subsatellites, Clementine, Lunar Prospector and other missions.

Retroreflector arrays were deployed by Apollo 11 in 1969, Lunokhod 1 in 1970, Apollo 14 and Apollo 15 in 1971, and Lunokhod 2 in 1973. Lunar laser ranging measurements began in 1969. The Lunokhod 1 reflector was effectively lost as a routine ranging target for decades until its location was rediscovered in Lunar Reconnaissance Orbiter imagery in 2010.

Round-trip light time between Earth and the Moon is about 2.5 seconds. Modern APOLLO measurements explicitly describe photons returning roughly 2.5 seconds after transmission.

Lunar recession is about 3.8 cm per year, as established through long-term lunar laser ranging and dynamical analysis.

Historical ILRS solutions show centimeter-scale RMS residuals for normal points. Modern APOLLO measurements provide millimeter-scale precision, with a median nightly range uncertainty of about 1.7 mm in the published 15-year dataset. Therefore, describing LLR simply as centimeter-scale undersells modern capability; historical centimeter-level and modern millimeter-level performance should be distinguished.

For GM, LLR is complementary to spacecraft tracking. Spacecraft tracking provides extremely strong constraints on the lunar gravity field and gravitational parameter, while LLR’s long time base provides powerful constraints on the Earth-Moon ephemeris, lunar rotation, tidal parameters and other aspects of lunar dynamics.