# Theoretical and Observational Constraints on Lunar Orbital Evolution in the Three-Body Earth-Moon-Sun System

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{−1}from modelling recent lunar laser ranging (LLR) measurements (e.g., [3,4,5,6]), is attributed to conversion of Earth’s spin, which is decreasing, into lunar orbital angular momentum, which is perceived as increasing (e.g., [7]). This transfer hypothesis approximates the Earth and Moon as being an isolated two-body system [8], a case where no torque can exist [9], so angular momentum of the system should be conserved.

- Torque is required to modify angular momentum [9], yet torque ($\tau =\overrightarrow{r}\times \overrightarrow{F}$) on the Moon originating from the Earth is zero, because their vectors for separation distance (r) and their attractive gravitational force (F) are collinear.
- No mechanism has been proposed for transforming axial spin of a body to the orbital angular momentum of a distant body.
- Equations for the gravitational origin of orbits do not depend on the spin of the central body: indeed, a body need not even exist at the system barycenter, as exemplified by Pluto and Charon.

**Figure 1.**Key features of the 3-body Earth-Moon-Sun system: (

**a**) Schematic of stable 3-body orbits around the stationary Sun in a plane fixed in space. Blue orb represents Earth. Large crescents depict stable orbital positions at the 60° and antipodal Lagrangian points. Tiny crescents depict quasi-stable points utilized for satellites; (

**b**) Quadrant of the lunar orbit showing a to scale comparison of the nearly circular barycenter orbit (solid line) to the Moon’s sinuous orbit (pink dots), which reaches a maximum of ~405,000 km from Earth (e.g., [10]). The actual excursion is slightly less than the maximum shown, due to the orbit being elliptical. The lunar monthly period is idealized as 1/12th of a year; (

**c**) Orientation of Earth’s spin and various orbital elements. The barycenter and ecliptic planes are distinct. Pink indicates components affected by the time-varying lunar orbit. The directions of the barycenter axis and the spin axis vary from ~0 to 28.5° over the lunar cycling, and precess independently. The radial position of the barycenter (B, pink double dots) varies by ~500 km over a month inside the lower mantle. Its longitude varies by 360° every day. Earth–Moon distance is not to scale, but body sizes and angles are. Part c was modified after [11], which has a Creative Commons 4 license.

## 2. Observations and Parameterization of the Unique Lunar Orbit and its Cycles

#### 2.1. Long-Standing Techniques for Measuring Motions in the Solar System

#### 2.2. Lunar Orbital Parameters and Velocities

Description | Symbol | Value | Description | Symbol | Value |
---|---|---|---|---|---|

Solar mass | M_{S} | 1.99 × 10^{30} kg | Gravitational constant | G | 6.674 × 10^{−11} m^{3} kg^{−1}s^{−2} |

Earth’s mass | M_{E} | 5.97 × 10^{24} kg | Barycenter orbit radius | r_{B} | 149.6 × 10^{6} km |

Lunar mass | M_{m} | 7.35 × 10^{22} kg | Lunar semi-major axis ^{2} | a | 384,748 km |

Earth’s average radius | R_{E} | 6371 km | Lunar eccentricity ^{2} | e | 0.05490 |

Moon’s average radius | R_{m} | 1738 km | Inclination to Earth’s equator ^{3} | i | 18–28° |

Earth’s ellipticity | ε | 0.003353 | Barycenter-geocenter distance | B | average ~4670 km |

^{1}Subscripts S refers to the Sun; E to the Earth, m to the Moon; and B for barycenter is used as both a symbol and subscript. Sources: [10,16,17]. Some apparent inconsistency among sources exists, as lunar orbital radii may refer to the geocenter, which is more convenient.

^{2}Refers to the Moon’s average ellipse around the barycenter, which varies monthly and yearly.

^{3}See Figure 1c for the many angles of components in the Earth-Moon-Sun system.

**Figure 2.**Geometry of the sinuous lunar orbit and its cycles: (

**a**) Plan view of the whirligig orbits of the Moon and Earth, and of the circular barycenter orbit, all about the Sun. The barycenter orbit is straight for simplicity. The crossings as sketched for the Moon and Earth are required for non-colliding paths and fixed orbital radius, but are distorted in this view; (

**b**) Plan view of an ideal 2-body orbit of the Moon around the Earth, for which representation, the Sun can be considered as a perturbing force. Hence, the Sun accelerates the Moon along (a

_{||}) and perpendicular (a

_{⊥}) to its orbit around the Earth except during the crossings of the barycenter orbit (black dots). The Moon is additionally accelerated towards the barycenter plane (a

_{tilt}at

**×**). Due to curvature of the barycenter orbit, the Moon spends more time outboard than inboard (stippled pie slice); (

**c**) Graph of days since 1 January 2001 between sequential first and last quarters of the Moon from January 2001 to the end of the 8.85 year apsidal precession cycle in 2009, obtained from tabulated data for the century [18], and confirmed by comparison to ephemeris tables [19]. This cycle describes a complete revolution of the orbital ellipse. The number of days outboard vs. inboard varies from −2.49 to +2.52, so the Moon spends a small excess fraction of the month outboard; (

**d**) Plan view of the orbits around the Sun, from a geocentric perspective, showing additional effects of the non-circular lunar orbit.

^{−1}for the Moon’s apparent path around the Earth, or ~30 km s

^{−1}for the barycenter around the Sun (e.g., [10]). To “keep pace with the Earth” the Moon moves faster when it is outboard of the barycenter, than inboard, but because the barycenter orbit is curved, the moon spends more time outboard (Figure 2).

#### 2.3. Lunar Cycles: “Year” vs. “Month”

#### 2.4. Motivation to Directly Measure the Moon’s Orbital Radius

^{−1}assuming zero radius at solar system beginnings. This upper limit for recession is 12 or 13 orders-of-magnitude smaller than orbital velocities (Section 2.2), and is far too small to quantify based on observations against the stars. Alternatively, as proposed by Laplace, the Moon could have formed from a ring of dust around the Earth (see, e.g., [23,24] and references therein). Co-accretion points to an originally circular orbit at some finite radius, permitting negative values for lunar drift. Section 5 provides details.

## 3. Gravitational Attraction to Oblate Planets and to the Pluto-Charon Binary

#### 3.1. Gravitational Attraction to Non-Spherical Mass Distributions

^{2}+ z

^{2})

^{½}. Moreover, at a general point the force is not even directed towards the object’s center (Figure 3b: see [25] for additional examples).

^{2}nor as 1/s

^{2}per Equations (1)–(6).

#### 3.1.1. Comparison with the Generalized Potential Used in Fitting

_{ave}is mean equatorial body radius; s, θ, and λ are the spherical orbital radius, latitude, and longitude of the test mass; P

_{n}

^{k}are associated Legendre functions of the first kind; and C

_{nk}and S

_{nk}are numerical coefficients obtained from fitting. As a potential for a planet must necessarily include their predominantly oblate shape, limitations of the generalized potential are understood by comparing Equation (7) to the exact results for an oblate body, Equations (1)–(6).

_{nk}and S

_{nk}terms involve longitude and k = 0, where the zonal harmonic coefficients (J

_{n}, defined as −C

_{n}

_{0}) are used in fitting, e.g., of motions of satellites around a gas giant. Equation (7) can only be reconciled with results for the special axes, Equations (1) and (2), if and only if the relationship for each term:

_{2}in particular, is lost in applying Equation (7) to the giant planets [25] which are distinctly oblate, rather than approximately spherical with bumps.

#### 3.1.2. Orbits involving A Non-Central Potential

#### 3.2. Satellite Systems Confirm That Earth Acts as A Point Mass on the Moon

#### 3.2.1. Examples of Orbits around Uniaxial Central Mass Distributions

#### 3.2.2. A Triaxially Shaped and Time-Varying Central Mass Distribution

#### 3.3. Ellipticity and Spin Rate

#### 3.3.1. Effect of Slowing Planet Spin on nearby Satellites

#### 3.3.2. Effect of Slowing Planet Spin on Distant Satellites

## 4. Internal Dissipation of Body Spin and Implications for Conservation Laws

#### 4.1. Oversimplified Case: Conservation of Spin Angular Momentum Holds

^{−1}relative to the mantle and below (~6300 km radius, sketched in Figure 6b). From a classical physics standpoint, friction exists between these layers and degrades the spin while heating their interface, but (angular) momentum is alleged to be conserved nevertheless. Given that the lithosphere is only 0.6 wt% of the Earth [36], recent slowdown of the surface at 2 × 10

^{−6}s y

^{−1}[37,38] means that if spin angular momentum were indeed conserved, then the interior would accelerate by a mere ~10

^{−9}s y

^{−1}.

^{−1}) makes quantification difficult (see [32] and the review of [39]). A tangential velocity similar to continental drift is well below the limit of detection. Notably, super-rotation of the whole core differs from super-rotation of the inner core. In particular, the molten outer core flows (shears) without resistance, so a faster rotating, more oblate solid inner core could be accommodated while the mantle rotates more slowly and is rounder. Criss [32] provides analytical formulae for spinning stratified bodies along with calculations for several endmember cases.

#### 4.2. Frictional Forces Decrease Spin

^{2}, where I is the moment of inertia). For additional discussion of large-scale processes occur inside the Earth, see [33]. Microscopic processes that dissipate momentum must be intimately tied with those converting mechanical energy to heat, but a detailed discussion is beyond the scope of this report.

## 5. Permissible Changes in Orbits around a Central Point Mass

#### 5.1. Conservation Laws for Orbits Are Stringent

#### 5.1.1. Energy Conservation Is Key

#### 5.1.2. Changes in the Lunar Orbit Permitted under Energy Conservation

_{tot}of an orbiting object does not permit changes in its semi-major axis, but allows eccentricity to vary [42]. Importantly, for a 2-body system, any inclination is permissible, as this factor does not enter into the description of the orbit around a point mass or sphere. Thus, orbital inclination may also change.

^{−1}. A circular orbit must describe the starting point because torque is elongating the orbit, as follows:

#### 5.2. Solar Torques Change Eccentriciy and Inclination of the Lunar Orbit

_{G,SB}= F

_{R,SB}, and similarly for a nearly stable lunar orbit: F

_{G,Em}≅ F

_{R,Em}(Figure 8). In contrast, F

_{G,Sm}and F

_{R,Sm}are imbalanced even in this idealization.

#### 5.2.1. Permissible Changes in Eccentricity

^{−1}while the minimum is 0.970 km s

^{−1}. Parallel acceleration should not change eccentricity. Orbital precession, i.e., rotation of the orbital ellipse as the barycenter moves around the Sun (Figure 2d), probably links to parallel acceleration.

^{−1}from Equation (14) and Figure 7b).

#### 5.2.2. Permissible Changes in Inclination

^{o}with respect of Earth’s equatorial plane (Figure 1c). The latter is relevant (Section 3.2). Assuming that an equatorial configuration existed at formation suggests a progression of ~5 × 10

^{−9}°y

^{−1}. Jupiter also applies torque, but to both Earth and its Moon, thereby moving the barycenter orbital plane (Section 7).

## 6. Uncertainties in Modelling Lunar Drift from LLR

#### 6.1. Characteristics of the Moving Lunar Target

#### 6.1.1. Daily Variations

^{−1}from perigee to apogee, and then shrinks in the second half of the month. Simultaneously, the distance from the barycenter to the geocenter changes proportionately:

^{−1}is nearly 10-orders of magnitude smaller than the average daily change in the lunar radius over a half-month (Figure 9). Accuracy in radius better than a parts-per-billion level is required to established drift. This level of precision is not achievable, as follows:

#### 6.1.2. Variations in Tangential Velocity during the Lunar Orbit

#### 6.2. Lunar Laser Ranging Data and How Drift Is Ascertained

^{−1}recession value is based. Satellite experiments began afterwards and so are not covered here. Models used to extract drift are discussed after we describe LLR raw data and compare these to distances ascertained from ephemeris tables [19,45,46].

#### 6.2.1. Available Station-to-Mirror Travel Times

^{−13}s, which corresponds to an uncertainty of ±0.03 mm for each uncorrected station-to-mirror distance.

#### 6.2.2. LLR Data Collection Is Too Infrequent to Describe Monthly and Longer Lunar Cycles

#### 6.2.3. Lunar Drift Is a Model Value

- The models focus on the lunar orbit around the geocenter, which requires utilizing parameterizations based on the lunar ephemerides [4,15]. Heliocentric orbital parameters of the Earth–Moon barycenter are also used [4]. Hence, due to under-sampling of the cycles, the radius determinations largely rest on astronomical observations, not on LLR acquisitions.
- An average value for the barycenter position is used [4], which is invalid when the station lies off of the line defined by the geocenter-barycenter-moon center. This station orientation commonly occurs as evidenced by LLR data overshooting near the apogee (Figure 12). These correction terms are hundreds of kilometers. In detail, the barycenter is not located at a fixed position inside the Earth, but varies longitudinally over the day, and radially over the anomalistic month (Figure 9), where the inclination of Earth’s spin axis to the lunar orbital plane (Figure 1), provides latitudinal variations.
- Muller et al. [7] mentions a global parameter adjustment where improved values of the unknowns and the corresponding formal standard errors are obtained. Adjusting the unknowns provides a false precision.
- Importantly, apparent agreement exists between LLR raw data and ephemeris tables roughly midway between perigee and apogee (Figure 12). This section of the orbit is unimportant because an elliptical orbit is defined by its apogee and perigee. The apsides are under-sampled by LLR, and are most sensitive to corrections. This serious problem is not discussed in available reports.
- Atmospheric effects, producing refraction, are highly variable. This poorly constrained correction term is circa 2 m, independent of all other modelling efforts.
- Last, but not least, elliptical orbits are described by two parameters: the semi-major axis and the eccentricity. Measurements of radius alone are insufficient, as two measurements are needed to solve for two unknowns. This serious problem has been overlooked.

## 7. Discussion

#### 7.1. The Lunar Orbital Radius Is Decreasing, Not Increasing

^{9}y after formation, remains nearly circular (Figure 7a), despite Solar forces acting on the Moon being double Earth’s. This observation testifies to stability of the lunar orbit. Conservation of energy, and Solar forces providing torque in the 3-body geometry cause eccentricity to increase, which contracts the semi-minor axis while maintaining the semi-major axis (Figure 7a). Consequently, the average lunar radius has decreased over geologic time (−0.13 mm y

^{−1}; Figure 7b). Inclination of the lunar orbit from Earth’s equatorial plane is consistent with Solar torque existing, unopposed, but the inclination is not part of the orbital energetics. The change in i is rather small and is consistent with stability and tiny changes in eccentricity.

#### 7.2. Comparative Planetology and Evolutionary Behavior throughout Our Solar System

^{−12}per year to 4.5 × 10

^{−11}which are similar to and bracket the Moon’s rate, established here as 1.2 × 10

^{−11}.

^{−8}°y

^{−1}, assuming an initially polar orbit, which is consistent with orientation of its current spin axis and with its equatorial satellite orbits. Assuming that the Moon’s orbit was in Earth’s equatorial plane at formation suggests a similar progression of ~0.5 × 10

^{−8}°y

^{−1}. Both Earth and its Moon are being pulled to the orbital plane of Jupiter around the Sun, as are Mercury, Venus, and Mars. However, much stronger forces from the Sun has caused wobbling of the lunar plane about the barycenter plane (Figure 1) which is still not parallel to Jupiter’s orbital plane.

#### 7.3. Implications for Formation in General and the Moon in Particular

- Large perturbing forces limit the applicability of the reduced 2-body approximation.
- Due to geometry, the fixed-plane three-body approximation is likewise insufficient to describe the Earth-Moon-Sun system.
- The highly variable 3-dimensional geometry of the lunar orbit permits the Sun to apply torque, which changes angular momentum.
- Forces on the Moon imperfectly balance, mainly due to the time-varying distance of the Moon from the Sun.
- However, without dissipation of orbital energy, which requires a non-conservative force such as friction, orbital evolution is limited to changes in eccentricity and inclination, as long as the 2-body approximation is reasonably accurate.

#### 7.4. Relationship of Our Work to Previous Studies

## 8. Conclusions

^{9}year) changes in orbits are not constrained by lunar laser ranging measurements (Section 6). Instead, we use conservation laws to deduce the following:

- The lunar orbital radius is shrinking, so drift is about −0.13 mm y
^{−1}. - The lunar orbit was originally circular and probably about Earth’s equator.
- Co-accretion is favored, if not proven, since these findings rule out all other contenders for the lunar beginnings (a giant impact, fission, or capture).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Gravitational forces around the upper right quadrant of a highly flattened oblate spheroid: (

**a**) Values of force along the two special directions for a body with ε/α = 0.1, shown as a grey oval (where ς is the polar radius), compared to the force around a sphere of equivalent mass (pink dotted curve). At 10 body radii (Distance = 10 α; offscale), the forces along special directions of this flattened spheroid only differ from than that of the sphere by ±0.3%; (

**b**) Directions of force, which are continuous from outside to inside the body (faint gray oval). Both parts modified after Hofmeister et al. [25] (Figures 4a and 7f therein), with permissions.

**Figure 4.**Orbital properties of moons of the solar system, showing correspondence of orbital inclination with respect to the planets’ equator with orbital eccentricity. Data from NASA [16]. Box links symbols to planetary systems. Least-squares fits include the origin: (

**a**) Plot differentiating retrograde moons. Large, round moons (α > 1400 km) are labelled. Only Triton is retrograde. Regarding tiny Hippocamp of Neptune, 5 moons of Saturn inside the main rings, and 3 moons inside Saturn’s ring E, zero was listed for either e or i or both. To depict these satellites on logarithmic plots, we set e = 8 × 10

^{−6}and/or i = 0.0008°, which values are below all reports for other moons: consequently these 9 moons plot on the axes or at “zero.” The main groupings, as labeled, suggest correlations with potential of the central body; (

**b**) Same data, except that retrograde moons are plotted with inclination reduced by 180

^{o}. Fits excluding retrograde moons are similar. Moons that deviate greatly from the depicted trends are labelled, as are all large moons.

**Figure 5.**Orbital eccentricity and inclination of moons as a function of their semi-major orbital axis normalized to the equatorial radius of the central planet, using scales appropriate to both distal and proximal moons. Data from NASA factsheets [16]. For the retrograde moons, the inclination shown = 180

^{o}minus that reported. Solid symbols = inclination (right axis) and open symbols = eccentricity (left axis), where the color scheme is the same as in Figure 4: (

**a**) Plot of all moons with a/α < 425. This cutoff excludes one of Saturn’s 79 moons which is near 450 body radii; plus 5 of Uranus’s 24 moons and 5 of Neptune’s 14 moons, all of which are much further out, reaching a/α = 10,000. These 11 moons orbit with a wide range of i and e, like the cluster near a/α of 300. Large Triton is retrograde in a tilted, circular orbit, whereas Hyperion has a chaotic orbit, attributed to interactions with Titan; (

**b**) Expanded view of the box near the origin in the left panel. Pluto’s tiny moons [30], which actually orbit the Pluto-Charon barycenter are not shown: see text for discussion of this unique satellite system.

**Figure 6.**Behavior of spinning self-gravitating bodyies: (

**a**) Spin of a homogeneous density, fully solidified oblate. Modified after Criss [32] (Figure 1 therein), with permissions; (

**b**) Primary mechanism of differential rotation for the Earth. Equatorial slice, showing internal structure, down the North pole (N), showing uniaxial stress cracks (black bars), layers (various patterns), and drag (black arrow). The Earth’s lower mantle (blue marble) spins through the barycenter (B) on a daily basis. This variation is not entirely longitudinal because the spin axis is inclined to the lunar orbit and to the barycenter orbit. Dark grey = liquid outer core. Samples are limited to origins in the upper mantle (light blue stipple). Modified after Hofmeister et al. [33] (Figure 2e therein), with permissions.

**Figure 7.**Geometry of the lunar orbit: (

**a**) Polar plot. Heavy black shows the current semi-major axis and eccentricity. The central dot shows the actual size of Earth. Black dotted curve is a circular orbit with the same energy. Additional permissible orbits obtained using Equation (13) are shown. Red long dashes show e = 0.206 which also describes Mercury’s orbit, the most eccentric among the planets. Green medium dashes = high e seen in orbits of some distant prograde moons (Figure 5). Purple short dashes = very high e, above which collision with Earth could occur; (

**b**) Calculation of the average lunar radius from its current semi-major axis and different values of e. Arrows show the changes associated with the limiting geometries of a circular orbit compared to a nearly linear orbit at 4.52 by ago.

**Figure 8.**Schematics of forces perpendicular to the barycenter plane associated with the reduced 2-body orbits in the Earth-Moon-Sun system. Object size and angles are exaggerated to reveal the perturbing forces and imbalances. Distances are not to scale. Forces are as labeled, with shade and length of filled-tip arrows suggesting relative magnitude. The imbalances, denoted Solar grab and sling, are shown as white open arrows. Sling involves the orbiting object not falling towards the center sufficiently fast for a stable orbit. Distances and/or orbits are indicated with various patterned lines: (

**a**) Cross section when the Moon is inside the barycenter orbit; (

**b**) Cross section when the Moon is outside the barycenter orbit.

**Figure 9.**Characteristics of the lunar orbit over the perigee-to-apogee half-cycle: (

**a**) Distances associated with the lunar and terrestrial Keplerian orbits around the barycenter. Red dots = lunar radii spaced at equal times of 0.54 days, which were calculated from Equation (13) using 0.2° increments, and a and e in Table 1. These points are reproduced by a 4th order polynomial fit (not shown). The average radius (black dashes) is slightly lower than the semi-major axis (grey heavy line). Blue X and right axis = distance to the geocenter, calculated from the fit; (

**b**) Time derivatives of the calculated distances. Red solid curve = lunar variations over the half-cycle. Blue dots near x axis = the barycenter variation on the same scale. Long dashes and right axis = lunar drift, as modelled from LLR delay times [3], scaled to a daily basis.

**Figure 10.**Histograms of the lunar Keplerian orbit over an anomalistic cycle, constructed for distances calculated for three uniform angular (~time) intervals. Panels have the same numbers of bins for direct comparison. Except for apogee and perigee, each degree increment provides 2 counts: (

**a**) Radius calculated at increments of 0.5°, which roughly corresponds with 0.038 day sampling intervals; (

**b**) Radius calculated at increments of 5°, which roughly corresponds with 0.38 day sampling intervals; (

**c**) Radius calculated at increments of 13°, which roughly corresponds with 1 day sampling intervals.

**Figure 11.**Histogram of the number of LLR measurements each year. Each bin is one year wide [ref for improved accuracy] Data from [3]. The last 2 years data were not reported, apparently due to the changeover to satellite stations. Horizontal lines show important cycles.

**Figure 12.**LLR (red dots) data [3] compared to the orbital radius from the ephemeris tables [45,46] less the combined body radii of the Moon and Earth (fine black line): (

**a**) Example of data collection at an average interval of ~1 day. Purple vertical bar shows the 8109 km body radii contribution, which is substantial; (

**b**) Example of data collected at an average interval near ½ day. Blue curve shows the monthly average distance. For both panels, ~90% of the data were collected at Grasse [7,44]. Note that distances longer than expected occur when the station is not collinear with the line containing the geocenter, barycenter, and Moon’s center.

**Figure 13.**Distances from travel times obtained from 1992 to 2001.5 reveal a “bowtie” pattern over the apsidal precession cycle (blue arrow). Black dots = individual measurements. Orange arrow shows evidence of yearly cycling. Red curve emphasizes the 8.85 year precession cycle connected with the minima. Green curve shows 8.85 year cycle as evident in the maxima. Pink shows the mean station-to-mirror distance (380,432 km) over a precession cycle.

Type | Size (mm) ^{2} | References |
---|---|---|

The center of Earth to center of Moon distance is computed by a computer program that numerically integrates the lunar and planetary orbits accounting for the gravitational attraction of the Sun, planets, and a selection of asteroids. ^{1} | See Section 6.2.3 | [3,5,6,49] |

Position of station, accounting for rock tides and seasonal motion of the solid Earth with respect to its center of mass. | 20 to 1000 | [3,49,51] |

Position of mirror, with respect to the lunar center (libration effects) | <1,000,000 | [3,49,51] |

Atmospheric refraction | ~2000 | [3,49,51] |

^{1}The latter is related to the location of the Solar system barycenter, which is noted by [5] and several others.

^{2}Size of correction terms mostly for Apache point [51]. This list does not include corrections that are small (e.g., thermal expansion of mirror mounts, relativity, or lag times for emission, reflection, and reception [6]).

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**MDPI and ACS Style**

Hofmeister, A.M.; Criss, R.E.; Criss, E.M.
Theoretical and Observational Constraints on Lunar Orbital Evolution in the Three-Body Earth-Moon-Sun System. *Astronomy* **2022**, *1*, 58-83.
https://doi.org/10.3390/astronomy1020007

**AMA Style**

Hofmeister AM, Criss RE, Criss EM.
Theoretical and Observational Constraints on Lunar Orbital Evolution in the Three-Body Earth-Moon-Sun System. *Astronomy*. 2022; 1(2):58-83.
https://doi.org/10.3390/astronomy1020007

**Chicago/Turabian Style**

Hofmeister, Anne M., Robert E. Criss, and Everett M. Criss.
2022. "Theoretical and Observational Constraints on Lunar Orbital Evolution in the Three-Body Earth-Moon-Sun System" *Astronomy* 1, no. 2: 58-83.
https://doi.org/10.3390/astronomy1020007