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Vol.3, No.5, 339-343 (2011) Natural Science http://dx.doi.org/10.4236/ns.2011.35045 Lunar deflections of the vertical and their distribution Jinyun Guo1,2*, Yu Sun1,2, Xiaotao Chang2,3, Fanlin Yang1,2 1College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao, China; *Corresponding Author: [email protected] 2Key Laboratory of Surveying and Mapping Technology on Island and Reef of SBSM, Qingdao, China; 3Satellite Surveying and Mapping Application Center of SBSM, Beijing, China. Received 4 February 2011; revised 20 March 2011; accepted 6 April 2011. ABSTRACT interior, from crust to core and further understanding of the thermal evolution of the Moon [1]. Lunar gravity The deflection of the vertical reflects the mass field, which contains abundant information related to distribution and density anomaly of celestial lunar internal mass distribution, plays a key role in orbit bodies. Lunar deflections of the vertical include determination and controlling lunar spacecrafts. directional information of the Moon’s gravity Abundant lunar gravity information can be extracted field. SGM90d, recovered from SELENE mission, from the ground tracking data of lunar satellite and ob- revealed the lunar far side gravity field for the servations from lunar explorers, which are scientific data first time in history owes to 4-way Doppler data. used to refine the lunar gravity field model [2]. Study of Lunar deflections of the vertical and their me- lunar gravity with ground tracking data began with the ridional and prime vertical components are cal- Soviet Union spacecraft Luna 10 lunched in 1966. Many culated from SGM90d, and then their global lunar probing missions have been implemented since distributions are also given in the paper. The then. Lunar Orbiter (LO-I, II, III, IV, and V), mini satel- gradients of lunar deflections of the vertical are lites of Apollo 15 and 16 (A15 and A16ss), Clementine, defined and computed as well. The correlations Lunar Prospector (LP), SMART-1, SELENE, ChangE between the lunar deflections of the vertical and and Chandrayaan provided plenty ground tracking data the lunar terrain have been fully discussed. Many to recovery lunar gravity field models. Observables from different characteristics of lunar deflections of these lunar missions include ranges, range rates, two-/th- vertical have been found between the near side ree-/four-way Doppler observations, D-VLBI data, di- and the far side of the Moon, which may be rection data and craft-borne laser altimetric terrain data. caused from the lithospheric compensation and These data include more frequences of lunar gravity the uplifting of mantle. field information [3-8]. Existing high-degree lunar grav- ity field models, such as LP100J, LP100K and LP165P, Keywords: Lunar Deflections of the Vertical; were estimated from tracking data of LP and historical Lunar Gravity Field; Lunar Topography; lunar crafts before SELENE mission was implemented Gradient of Vertical Deflection [5,9,10]. It is a known fact that there are no direct ob- servations of the farside, and the observing precision at 1. INTRODUCTION the Moon edge is low. Meantime the distribution of tracking data is not even and there are more data at the The Moon is the natural celestial body closest to our lunar equatorial area. So these lunar gravity models have living planet, Earth. There are many fantasies about the much higher resolution at the nearside and lower resolu- Moon since it could be the second home for human. tion at the farside. Also, there are more errors and alias- Many lunar probing missions had been implemented by ing at the truncated degrees for these models [5]. SE- the Soviet Union, the United States, Europe, Japan, LENE can directly be tracked at the nearside and the China and India since 1960s. Depending on mission ob- farside from the earth stations, and the two-/four-way jectives, lunar spacecrafts were either used to detect Doppler data and ranges can be collected [11,12]. A new moon on a certain orbit or impacted on the surface of the lunar gravity field model SGM90d to degree and order Moon at last or soft landed on it with or without astro- of 90 has been developed from the SELENE-tracking nauts. The primary science objectives of lunar explora- data and historical data [13]. SGM90d is the first model tion including determination of the structure of the lunar including the direct observations at the farside. The lunar Copyright © 2011 SciRes. OPEN ACCESS 340 J. Y. Guo et al. / Natural Science 3 (2011) 339-343 gravity field model is mainly used to study the lunar degree and order of the gravity field model; Cnm and air-free gravity anomalies and the Bouguer anomalies, Snm the fully normalized lunar potential coefficients; and discuss the lunar gravity anomalies at mascons and Pnm sin the fully normalized associated Legendre basins. function, and the azimuth of the vertical deflection. The deflection of the vertical is an angle between the Namiki et al. have developed a new lunar gravity field direction of gravity and a reference direction [14]. Lunar model up to degree and order of 90 named SGM90d. deflections of the vertical indicate the slope of selenoid Tracking data used to determine this gravity model in- relative to the reference lunar ellipsoid, as well as the cludes SELENE tracking data from Oct. 31, 2007 to angle between the practical lunar vertical and the normal Apr. 1, 2008 as well as historical lunar satellites tracking lunar gravity direction. So the lunar deflections of the data [13]. Though has a lower degree and order, vertical include information of internal lunar mass dis- SGM90d reveals lunar far side ring shaped gravity fea- tribution and anomaly. Lunar gravity vector is composed tures in a much higher resolution than existing lunar of the vertical deflection and gravity anomaly based on a gravity models like LP100k. This significant improve- reference normal gravity. Lunar deflection of the vertical, ment owes to SELENE 4way Doppler observations, as one basic observation in the lunar geophysics and which enables direct tracking of the satellite over far selenodesy, can provide abundant information of lunar side of the Moon. The deflection of vertical and its me- gravity field and selenoid. So the lunar deflections of the ridional and prime vertical components and direction are vertical are very important for the study of lunar gravity calculated from SGM90d, drew with GMT [16], shown field. But we cannot directly precisely measure the ver- in Figure 1. tical deflections on the lunar surface at present. Lunar The global distribution of meridional components of deflections of the vertical and its distribution are calcu- lunar vertical deflections has been shown in Figure 1(a). lated based on SGM90d in this paper. Then the correla- Impacted basins locating at the near side including Im- tion between lunar deflections of the vertical and lunar brium, Serentitatis, Crisium and Humorum have similar topography is analyzed. Differences of vertical deflec- symmetrical patterns with positive meridional compo- tions on lunar mascons and basins are also discussed. nents at east and negative ones at west. Nectaris and Smythii have rather complicated distribution patterns, 2. LUNAR DEFLECTIONS OF THE which are positive-negative-positive-negative (from west VERTICAL to east). Orientale, Hertzsprung, Freundlich-Sharonov, According to the definition of Molodensky deflection Moscoviense, Apollo and Mendel-Rydberg locating at of the vertical [15], lunar deflection of the vertical can the farside, on the other hand, have negative-positive-ne- be calculated based on the lunar gravity model from the gative-positive (from west to east) distribution patterns. following formulas: The global distribution of prime vertical components n presented in Figure 1(b) also shows similar symmetrical GMN a 2 pattern, but with a different symmetry axis direction. r n2 r (1) Four impacted basins, Imbrium, Serenitatis, Crisium and n dsinP Nectaris, have positive prime vertical components at CmSmcos sin nm nm nm northern part of the basins and negative ones on the oth- m0 d er haft. Humorum, Orientale and Mendel-Rydberg have n GMN a negative-positive-negative-positive distribution pat- terns 2 r cos r in north-south direction. n2 (2) n From the global distribution of lunar deflections of the mCnmsin m S nm cos mP nm sin vertical (shown in Figure 1(c)), ring shaped distribution m0 patterns can be fond at impacted basins including Im- brium, Serenitatis, Crisium, Nectaris and Humorum. VD 22 (3) Vertical deflections are very large at the periphery of the basins and much smaller in the center area. The circles tan (4) formed by large deflections of the vertical closely related to the lunar terrain and clearly indicate the boundaries of where and the meridional and the prime vertical original mascons [13]. The large deflections of the ver- components, respectively; VD the deflection of the ver- tical at Orientale formed two concentric circles. The tical; GM the lunar gravitational constant; the nor- outer ring is by the edge of the basin and inner one mal lunar gravity; r,, the spherical coordinates, seems to be the margin of gravity anomaly in the basin. N the highest degree of the model used; n and m And it is the same case for other basins like Hertzsprung, Copyright © 2011 SciRes. OPEN ACCESS J. Y. Guo et al. / Natural Science 3 (2011) 339-343 341 (a) (b) (c) (d) Figure 1. Distribution of lunar vertical deflections: (a) meridional, (b) prime vertical, (c) deflection of the vertical and (d) azimuth; the X-coordinate represents longitude and the Y-coordinate represents latitude. Lorentz, Korolev, Apollo, Freundlich-Sharonov and Mos- So the gradient of lunar vertical deflection can be calcu- coviense. Lorentz and Korolev have same air-free grav- lated with SGM90d, shown in Figure 2.
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