사용자:Aspere/달의 중력

표고에 따른 지구, 화성, 달의 중력 크기 비교.

달 표면에서의 방사상 중력 이상 (mGal).

달 표면에서의 중력 가속도는 약 1.625 m/s²로, 지구의 약 16.6%, 0.166 ɡ이다. 표면 전체에서의 중력 가속도 변동치는 약 0.0253 m/s²(중력 가속도의 약 1.6%)이다. 무게는 중력 가속도와 직접적인 관계가 있기 때문에, 달에서는 지구에 비해 물체의 무게가 약 6분의 1(16.6%)만 나간다.

중력장

달의 중력장은 달을 공전하는 우주선이 중력의 차이로 인해 발생하는 미세한 가속도를 우주선에서 방출한 전파의 도플러 효과로 측정하는 방법을 이용하여 구할 수 있으며, 이를 종합하면 달의 중력이상까지 알아낼 수 있다.

중력장의 차이로 인해 달에서 저궤도 대부분은 불안정하며, 실제 측정 결과 안전하게 유지되는 달 저궤도는 궤도 경사가 27°, 50°, 76°, 86°가 되는 지점밖에 없다. 달의 동주기 자전으로 인해 지구에서 달의 뒷면에 있는 우주선을 직접적으로 추적할 수 없기 때문에, 그레일 탐사선 전까지는 뒷면의 중력장을 측정하지 못하였다.

달 표면에서의 중력 가속도(m/s²). 앞면이 왼쪽, 뒷면이 오른쪽이다.

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.

달 중력 측정에 사용한 탐사선

임무명	ID	탐사선 개수	국가	연도
루나 오비터 1호	LO1	1	미국	1966
루나 오비터 2호	LO2	1	미국	1966–1967
루나 오비터 3호	LO3	1	미국	1967
루나 오비터 4호	LO4	1	미국	1967
루나 오비터 5호	LO5	1	미국	1967–1968
아폴로 15호 소형 인공위성	A15	1	미국	1971–1972
아폴로 16호 소형 인공위성	A16	1	미국	1972
클레멘타인	Cl	1	미국	1994
루나 프로스펙터	LP	1	미국	1998–1999
월주회위성 카구야	K/S	3	일본	2007–2009
창어 1호	Ch1	1	중국	2007–2009
그레일	G	2	미국	2012
창어 5호 T1	Ch1T1	1	중국	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
GLGM3	150	LO A15 A16 Cl LP
CEGM01	50	Ch 1
SGM100h	100	LO A15 A16 Cl LP K/S
SGM150J	150	LO A15 A16 Cl LP K/S
CEGM02	100	LO A15 A16 Cl LP K/S Ch1
GL0420A	420	G
GL0660B	660	G
GRGM660PRIM	660	G
GL0900D	900	G
GRGM900C	900	G
GRGM1200A	1200	G
CEGM03	100	LO A15 A16 Cl LP Ch1 K/S Ch5T1

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: 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. 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. 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.

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).

달의 질량

일반적으로 중력 상수 G는 질량과 중력 상수의 곱인 표준 중력 변수(μ = GM)보다 측정의 정확도가 낮다. 이로 인해 일반적으로 달의 질량을 표시할 때는 곱인 GM 형태로 나타낸다. 그레일 탐사선의 측정 결과, 달에서의 GM = 4902.8001 km³/s²이며, 이를 통해 구한 달의 질량 M = 7.3458 × 10²² kg이고, 평균 밀도는 3346 kg/m³이다. 달의 GM은 지구의 GM 값의 약 1/81.30057이다.

Theory

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 P_nm. 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,

$$V=\left({\frac {GM}{r}}\right)-\left({\frac {GM}{r}}\right)\sum \left({\frac {R}{r}}\right)^{n}J_{n}P_{n,0}(\sin \phi )+\left({\frac {GM}{r}}\right)\sum \left({\frac {R}{r}}\right)^{n}[C_{n,m}P_{n,m}(\sin \phi )\cos(m\lambda )+S_{n,m}P_{n,m}(\sin \phi )\sin(m\lambda )]$$

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 P_nm can be normalized or unnormalized affecting the gravity coefficients J_n, C_nm, and S_nm. Here we will use unnormalized functions and compatible coefficients. The P_n0 are called Legendre polynomials and the P_nm 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

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

Note that of the three functions, only P₂₀(±1)=1 is finite at the poles. More generally, only P_n0(±1)=1 are finite at the poles.

The gravitational acceleration of vector position r is

$${\begin{aligned}{\frac {d^{2}r}{dt^{2}}}&=\nabla V\\[1ex]&={\partial V \over \partial r}e_{r}+{\frac {1}{r}}{\partial V \over \partial \phi }e_{\phi }+{\frac {1}{r\cos \phi }}{\partial V \over \partial \lambda }{e_{\lambda }}\end{aligned}}$$

where e_r, e_φ, and e_λ are unit vectors in the three directions.

Gravity coefficients

The unnormalized gravity coefficients of degree 2 and 3 that were determined by the GRAIL mission are given in Table 1. 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

 Artificial gravity § Simulating lunar gravity 문서를 참고하십시오.

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. 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.

같이 보기

• 달의 자기장
• 무중량상태

각주

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