Gravitation of the Moon

Earth vs Mars vs Moon gravity at elevation Radial gravity anomaly at the surface of the Moon in mGal

The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 ɡ.[1] Over the entire surface, the variation in gravitational acceleration is about 0.0253 m/s2 (1.6% of the acceleration due to gravity). Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6% (= 1/6) of what they weigh on the Earth.

## Gravitational field

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The gravitational field of the Moon has been measured by tracking the radio signals emitted by orbiting spacecraft. The principle used depends on the Doppler effect, whereby the line-of-sight spacecraft acceleration can be measured by small shifts in frequency of the radio signal, and the measurement of the distance from the spacecraft to a station on Earth. Since the gravitational field of the Moon affects the orbit of a spacecraft, one can use this tracking data to detect gravity anomalies.

Most low lunar orbits are unstable. Detailed data collected has shown that for low lunar orbit the only "stable" orbits are at inclinations near 27°, 50°, 76°, and 86°.[2] Because of the Moon's synchronous rotation it is not possible to track spacecraft from Earth much beyond the limbs of the Moon, so until the recent Gravity Recovery and Interior Laboratory (GRAIL) mission the far-side gravity field was not well mapped.

Gravity acceleration at the surface of the Moon in m/s2. Near side on the left, far side on the right. Map from Lunar Gravity Model 2011.[3]

The missions with accurate Doppler tracking that have been used for deriving gravity fields are in the accompanying table. The table gives the mission spacecraft name, a brief designation, the number of mission spacecraft with accurate tracking, the country of origin, and the time span of the Doppler data. Apollos 15 and 16 released subsatellites. The Kaguya/SELENE mission had tracking between 3 satellites to get far-side tracking. GRAIL had very accurate tracking between 2 spacecraft and tracking from Earth.

Missions Used for Lunar Gravity   Mission  | ID  | Number  | Source  | Years

|
|
|
|

Lunar Orbiter 1 | LO1  | 1  | US  | 1966
Lunar Orbiter 2 | LO2  | 1  | US  | 1966–1967
Lunar Orbiter 3 | LO3  | 1  | US  | 1967
Lunar Orbiter 4 | LO4  | 1  | US  | 1967
Lunar Orbiter 5 | LO5  | 1  | US  | 1967–1968
Apollo 15 Subsatellite  | A15  | 1  | US  | 1971–1972
Apollo 16 Subsatellite  | A16  | 1  | US  | 1972
Clementine | Cl  | 1  | US  | 1994
Lunar Prospector | LP  | 1  | US  | 1998–1999
Kaguya/SELENE | K/S  | 3  | Japan  | 2007–2009
Chang’e 1 | Ch1  | 1  | China  | 2007–2009
GRAIL | G  | 2  | US  | 2012
Chang’e 5T1 | Ch1T1  | 1  | China  | 2015–2018

The accompanying table below lists lunar gravity fields. The table lists the designation of the gravity field, the highest degree and order, a list of mission IDs that were analyzed together, and a citation. Mission ID LO includes all 5 Lunar Orbiter missions. The GRAIL fields are very accurate; other missions are not combined with GRAIL.

Lunar Gravity Fields  Designation  | Degree  | Mission IDs  | Citation

|
|
|

LP165P  | 165  | LO A15 A16 Cl LP  | [4]
GLGM3  | 150  | LO A15 A16 Cl LP  | [5]
CEGM01  | 50  | Ch 1  | [6]
SGM100h  | 100  | LO A15 A16 Cl LP K/S  | [7]
SGM150J  | 150  | LO A15 A16 Cl LP K/S  | [8]
CEGM02  | 100  | LO A15 A16 Cl LP K/S Ch1  | [9]
GL0420A  | 420  | G  | [10]
GL0660B  | 660  | G  | [11]
GRGM660PRIM  | 660  | G  | [12]
GL0900D  | 900  | G  | [13]
GRGM900C  | 900  | G  | [14]
GRGM1200A  | 1200  | G  | [15]
CEGM03  | 100  | LO A15 A16 Cl LP Ch1 K/S Ch5T1  | [16]

A major feature of the Moon's gravitational field is the presence of mascons, which are large positive gravity anomalies associated with some of the giant impact basins. These anomalies significantly influence the orbit of spacecraft around the Moon, and an accurate gravitational model is necessary in the planning of both crewed and uncrewed missions. They were initially discovered by the analysis of Lunar Orbiter tracking data:[17] navigation tests prior to the Apollo program showed positioning errors much larger than mission specifications.

Mascons are in part due to the presence of dense mare basaltic lava flows that fill some of the impact basins.[18] However, lava flows by themselves cannot fully explain the gravitational variations, and uplift of the crust-mantle interface is required as well. Based on Lunar Prospector gravitational models, it has been suggested that some mascons exist that do not show evidence for mare basaltic volcanism.[4] The huge expanse of mare basaltic volcanism associated with Oceanus Procellarum does not cause a positive gravity anomaly. The center of gravity of the Moon does not coincide exactly with its geometric center, but is displaced toward the Earth by about 2 kilometers.[19]

Moon – Oceanus Procellarum ("Ocean of Storms")

Ancient rift valleys – rectangular structure (visible – topography – GRAIL gravity gradients) (October 1, 2014).

Ancient rift valleys – context.

Ancient rift valleys – closeup (artist's concept).

## Mass of Moon

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The gravitational constant G is less accurate than the product of G and masses for Earth and Moon. Consequently, it is conventional to express the lunar mass M multiplied by the gravitational constant G. The lunar GM = 4902.8001 km3/s2 from GRAIL analyses.[13][12][20] The mass of the Moon is M = 7.3458 × 1022 kg and the mean density is 3346 kg/m3. The lunar GM is 1/81.30057 of the Earth's GM.[21]

## Theory

[edit]

For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. The gravity potential is written with a series of spherical harmonic functions Pnm. The gravitational potential V at an external point is conventionally expressed as positive in astronomy and geophysics, but negative in physics. Then, with the former sign,

V = ( G M r ) − ( G M r ) ∑ ( R r ) n J n P n , 0 ( sin ⁡ ϕ ) + ( G M r ) ∑ ( R r ) n [ C n , m P n , m ( sin ⁡ ϕ ) cos ⁡ ( m λ ) + S n , m P n , m ( sin ⁡ ϕ ) sin ⁡ ( m λ ) ] {\displaystyle V=\left({\frac {GM}{r}}\right)-\left({\frac {GM}{r}}\right)\sum \left({\frac {R}{r}}\right)^{n}J_{n}P_{n,0}(\sin \phi )+\left({\frac {GM}{r}}\right)\sum \left({\frac {R}{r}}\right)^{n}[C_{n,m}P_{n,m}(\sin \phi )\cos(m\lambda )+S_{n,m}P_{n,m}(\sin \phi )\sin(m\lambda )]}

where r is the radius to an external point with r ≥ R, φ is the latitude of the external point, and λ is the east longitude of the external point. Note that the spherical harmonic functions Pnm can be normalized or unnormalized affecting the gravity coefficients Jn, Cnm, and Snm. Here we will use unnormalized functions and compatible coefficients. The Pn0 are called Legendre polynomials and the Pnm with m≠0 are called the Associated Legendre polynomials, where subscript n is the degree, m is the order, and m ≤ n. The sums start at n = 2. The unnormalized degree-2 functions are

P 2 , 0 = 3 2 sin 2 ϕ − 1 2 P 2 , 1 = 3 sin ⁡ ϕ cos ⁡ ϕ P 2 , 2 = 3 cos 2 ⁡ ϕ {\displaystyle {\begin{aligned}P_{2,0}&={\frac {3}{2}}\sin ^{2}\\!\phi -{\frac {1}{2}}\\\\[1ex]P_{2,1}&=3\sin \phi \cos \phi \\\\[1ex]P_{2,2}&=3\cos ^{2}\phi \end{aligned}}}

Note that of the three functions, only P20(±1)=1 is finite at the poles. More generally, only Pn0(±1)=1 are finite at the poles.

The gravitational acceleration of vector position r is

d 2 r d t 2 = ∇ V = ∂ V ∂ r e r + 1 r ∂ V ∂ ϕ e ϕ + 1 r cos ⁡ ϕ ∂ V ∂ λ e λ {\displaystyle {\begin{aligned}{\frac {d^{2}r}{dt^{2}}}&=\nabla V\\\\[1ex]&={\partial V \over \partial r}e_{r}+{\frac {1}{r}}{\partial V \over \partial \phi }e_{\phi }+{\frac {1}{r\cos \phi }}{\partial V \over \partial \lambda }{e_{\lambda }}\end{aligned}}}

where er, eφ, and eλ are unit vectors in the three directions.

## Gravity coefficients

[edit]

The unnormalized gravity coefficients of degree 2 and 3 that were determined by the GRAIL mission are given in Table 1.[13][12][20] The zero values of C21, S21, and S22 are because a principal axis frame is being used. There are no degree-1 coefficients when the three axes are centered on the center of mass.

Lunar Gravity Coefficients  nm | Jn | Cnm | Snm

|
|
|

20  | 203.3 × 10−6 | —  | —
21  | —  | 0  | 0
22  | —  | 22.4 × 10−6 | 0
30  | 8.46 × 10−6 | —  | —
31  | —  | 28.48 × 10−6 | 5.89 × 10−6
32  | —  | 4.84 × 10−6 | 1.67 × 10−6
33  | —  | 1.71 × 10−6 | −0.25 × 10−6

The J2 coefficient for an oblate shape to the gravity field is affected by rotation and solid-body tides whereas C22 is affected by solid-body tides. Both are larger than their equilibrium values showing that the upper layers of the Moon are strong enough to support elastic stress. The C31 coefficient is large.

## Simulating lunar gravity

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See also: Artificial gravity § Simulating lunar gravity

In January 2022 China was reported by the South China Morning Post to have built a small (60 centimeters in diameter) research facility to simulate low lunar gravity with the help of magnets.[22][23] The facility was reportedly partly inspired by the work of Andre Geim (who later shared the 2010 Nobel Prize in Physics for his research on graphene) and Michael Berry, who both shared the Ig Nobel Prize in Physics in 2000 for the magnetic levitation of a frog.[22][23]

## See also

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* Solar System portal

* Magnetic field of the Moon
* Micro-g environment

## References

[edit]

1. ^ C. Hirt; W. E. Featherstone (2012). "A 1.5 km-resolution gravity field model of the Moon". Earth and Planetary Science Letters. 329–330: 22–30\. Bibcode:2012E&PSL.329...22H. doi:10.1016/j.epsl.2012.02.012. Retrieved 2012-08-21.
2. ^ Bell, Trudy E. (November 6, 2006). Phillips, Tony (ed.). "Bizarre Lunar Orbits". Science@NASA. NASA. Archived from the original on 2021-12-04. Retrieved 2017-09-08.
3. ^ Lunar Gravity Model 2011, Archived 2013-01-14 at the Wayback Machine.
4. ^ a b A. Konopliv; S. Asmar; E. Carranza; W. Sjogren; D. Yuan (2001). "Recent gravity models as a result of the Lunar Prospector mission". Icarus. 50 (1): 1–18\. Bibcode:2001Icar..150....1K. CiteSeerX 10.1.1.18.1930. doi:10.1006/icar.2000.6573.
5. ^ Mazarico, E.; Lemoine, F. G.; Han, Shin-Chan; Smith, D. E. (2010). "GLGM-3: A degree-150 lunar gravity model from the historical tracking data of NASA Moon orbiters". Journal of Geophysical Research. 115 (E5): E05001, 1–14\. Bibcode:2010JGRE..115.5001M. doi:10.1029/2009JE003472. ISSN 0148-0227.
6. ^ Jianguo, Yan; Jinsong, Ping; Fei, Li; Jianfeng, Cao; Qian, Huang; Lihe, Fung (2010). "Chang'E-1 precision orbit determination and lunar gravity field solution". Advances in Space Research. 46 (1): 50–57\. Bibcode:2010AdSpR..46...50J. doi:10.1016/j.asr.2010.03.002.
7. ^ Matsumoto, K.; Goossens, S.; Ishihara, Y.; Liu, Q.; Kikuchi, F.; Iwata, T.; Namiki, N.; Noda, H.; Hanada, H.; et al. (2010). "An improved lunar gravity field model from SELENE and historical tracking data: Revealing the farside gravity features". Journal of Geophysical Research. 115 (E6): E06007, 1–20\. Bibcode:2010JGRE..115.6007M. doi:10.1029/2009JE003499. ISSN 0148-0227.
8. ^ Mazarico, E.; Lemoine, F. G.; Han, Shin-Chan; Smith, D. E. (2010). "GLGM-3: A degree-150 lunar gravity model from the historical tracking data of NASA Moon orbiters". Journal of Geophysical Research. 115 (E5): E05001, 1–14\. Bibcode:2010JGRE..115.5001M. doi:10.1029/2009JE003472. ISSN 0148-0227.
9. ^ Yan, Jianguo; Goossens, Sander; Matsumoto, Koji; Ping, Jinsong; Harada, Yuji; Iwata, Takahiro; Namiki, Noriyuki; Li, Fei; Tang, Geshi; et al. (2012). "CEGM02: An improved lunar gravity model using Chang'E-1 orbital tracking data". Planetary and Space Science. 62 (1): 1–9\. Bibcode:2012P&SS...62....1Y. doi:10.1016/j.pss.2011.11.010.
10. ^ Zuber, M. T.; Smith, D. E.; Neumann, G. A.; Goossens, S.; Andrews-Hanna, J. C.; Head, J. W.; Kiefer, W. S.; Asmar, S. W.; Konopliv, A. S.; et al. (2016). "Gravity field of the Orientale basin from the Gravity Recovery and Interior Laboratory Mission". Science. 354 (6311): 438–441\. Bibcode:2016Sci...354..438Z. doi:10.1126/science.aag0519. ISSN 0036-8075. PMC 7462089. PMID 27789835.
11. ^ Konopliv, Alex S.; Park, Ryan S.; Yuan, Dah-Ning; Asmar, Sami W.; Watkins, Michael M.; Williams, James G.; Fahnestock, Eugene; Kruizinga, Gerhard; Paik, Meegyeong; et al. (2013). "The JPL lunar gravity field to spherical harmonic degree 660 from the GRAIL Primary Mission". Journal of Geophysical Research: Planets. 118 (7): 1415–1434\. Bibcode:2013JGRE..118.1415K. doi:10.1002/jgre.20097. hdl:1721.1/85858. S2CID 16559256.
12. ^ a b c Lemoine, Frank G.; Goossens, Sander; Sabaka, Terence J.; Nicholas, Joseph B.; Mazarico, Erwan; Rowlands, David D.; Loomis, Bryant D.; Chinn, Douglas S.; Caprette, Douglas S.; Neumann, Gregory A.; Smith, David E. (2013). "High‒degree gravity models from GRAIL primary mission data". Journal of Geophysical Research: Planets. 118 (8): 1676–1698\. Bibcode:2013JGRE..118.1676L. doi:10.1002/jgre.20118. hdl:2060/20140010292. ISSN 2169-9097.
13. ^ a b c Konopliv, Alex S.; Park, Ryan S.; Yuan, Dah-Ning; Asmar, Sami W.; Watkins, Michael M.; Williams, James G.; Fahnestock, Eugene; Kruizinga, Gerhard; Paik, Meegyeong; Strekalov, Dmitry; Harvey, Nate (2014). "High-resolution lunar gravity fields from the GRAIL Primary and Extended Missions". Geophysical Research Letters. 41 (5): 1452–1458\. Bibcode:2014GeoRL..41.1452K. doi:10.1002/2013GL059066.
14. ^ Lemoine, Frank G.; Goossens, Sander; Sabaka, Terence J.; Nicholas, Joseph B.; Mazarico, Erwan; Rowlands, David D.; Loomis, Bryant D.; Chinn, Douglas S.; Neumann, Gregory A.; Smith, David E.; Zuber, Maria T. (2014). "GRGM900C: A degree 900 lunar gravity model from GRAIL primary and extended mission data". Geophysical Research Letters. 41 (10): 3382–3389\. Bibcode:2014GeoRL..41.3382L. doi:10.1002/2014GL060027. ISSN 0094-8276. PMC 4459205. PMID 26074638.
15. ^ Goossens, Sander; et, al. (2016). "A global degree and order 1200 model of the lunar gravity field using GRAIL mission data" (PDF).
16. ^ Yan, Jianguo; Liu, Shanhong; Xiao, Chi; Ye, Mao; Cao, Jianfeng; Harada, Yuji; Li, Fei; Li, Xie; Barriot, Jean-Pierre (2020). "A degree-100 lunar gravity model from the Chang'e 5T1 mission". Astronomy & Astrophysics. 636: A45, 1–11\. Bibcode:2020A&A...636A..45Y. doi:10.1051/0004-6361/201936802. ISSN 0004-6361. S2CID 216482920.
17. ^ P. Muller; W. Sjogren (1968). "Mascons: Lunar mass concentrations". Science. 161 (3842): 680–84\. Bibcode:1968Sci...161..680M. doi:10.1126/science.161.3842.680. PMID 17801458. S2CID 40110502.
18. ^ Richard A. Kerr (12 April 2013). "The Mystery of Our Moon's Gravitational Bumps Solved?". Science. 340 (6129): 138–39\. doi:10.1126/science.340.6129.138-a. PMID 23580504.
19. ^ Nine Planets
20. ^ a b Williams, James G.; Konopliv, Alexander S.; Boggs, Dale H.; Park, Ryan S.; Yuan, Dah-Ning; Lemoine, Frank G.; Goossens, Sander; Mazarico, Erwan; Nimmo, Francis; Weber, Renee C.; Asmar, Sami W. (2014). "Lunar interior properties from the GRAIL mission". Journal of Geophysical Research: Planets. 119 (7): 1546–1578\. Bibcode:2014JGRE..119.1546W. doi:10.1002/2013JE004559. S2CID 7045590.
21. ^ Park, Ryan S.; Folkner, William M.; Williams, James G.; Boggs, Dale H. (2021). "The JPL Planetary and Lunar Ephemerides DE440 and DE441". The Astronomical Journal. 161 (3): 105. Bibcode:2021AJ....161..105P. doi:10.3847/1538-3881/abd414. ISSN 1538-3881. S2CID 233943954.
22. ^ a b "China building "Artificial Moon" that simulates low gravity with magnets". Futurism.com. Recurrent Ventures. Retrieved 17 January 2022. "Interestingly, the facility was partly inspired by previous research conducted by Russian physicist Andrew Geim in which he floated a frog with a magnet. The experiment earned Geim the Ig Nobel Prize in Physics, a satirical award given to unusual scientific research. It's cool that a quirky experiment involving floating a frog could lead to something approaching an honest-to-God antigravity chamber."
23. ^ a b Stephen Chen (12 January 2022). "China has built an artificial moon that simulates low-gravity conditions on Earth". South China Morning Post. Retrieved 17 January 2022. "It is said to be the first of its kind and could play a key role in the country's future lunar missions. Landscape is supported by a magnetic field and was inspired by experiments to levitate a frog."

* v
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Moon

Outline

Physical
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Orbit|

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