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:
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
The deflection of the vertical reflects the mass
distribution and density anomaly of celestial
bodies. Lunar deflections of the vertical include
directional information of the Moon’s gravity
field. SGM90d, recovered from SELENE mission,
revealed the lunar far side gravity field for the
first time in history owes to 4-way Doppler data.
Lunar deflections of the vertical and their me-
ridional and prime vertical components are cal-
culated from SGM90d, and then their global
distributions are also given in the paper. The
gradients of lunar deflections of the vertical are
defined and computed as well. The correlations
between the lunar deflections of the vertical and
the lunar terrain have been fully discussed. Many
different characteristics of lunar deflections of
vertical have been found between the near side
and the far side of the Moon, which may be
caused from the lithospheric compensation and
the uplifting of mantle.
Keywords: Lunar Deflections of the Vertical;
Lunar Gravity Field; Lunar Topography;
Gradient of Vertical Deflection
1. INTRODUCTION
The Moon is the natural celestial body closest to our
living planet, Earth. There are many fantasies about the
Moon since it could be the second home for human.
Many lunar probing missions had been implemented by
the Soviet Union, the United States, Europe, Japan,
China and India since 1960s. Depending on mission ob-
jectives, lunar spacecrafts were either used to detect
moon on a certain orbit or impacted on the surface of the
Moon at last or soft landed on it with or without astro-
nauts. The primary science objectives of lunar explora-
tion including determination of the structure of the lunar
interior, from crust to core and further understanding of
the thermal evolution of the Moon [1]. Lunar gravity
field, which contains abundant information related to
lunar internal mass distribution, plays a key role in orbit
determination and controlling lunar spacecrafts.
Abundant lunar gravity information can be extracted
from the ground tracking data of lunar satellite and ob-
servations from lunar explorers, which are scientific data
used to refine the lunar gravity field model [2]. Study of
lunar gravity with ground tracking data began with the
Soviet Union spacecraft Luna 10 lunched in 1966. Many
lunar probing missions have been implemented since
then. Lunar Orbiter (LO-I, II, III, IV, and V), mini satel-
lites of Apollo 15 and 16 (A15 and A16ss), Clementine,
Lunar Prospector (LP), SMART-1, SELENE, ChangE
and Chandrayaan provided plenty ground tracking data
to recovery lunar gravity field models. Observables from
these lunar missions include ranges, range rates, two-/th-
ree-/four-way Doppler observations, D-VLBI data, di-
rection data and craft-borne laser altimetric terrain data.
These data include more frequences of lunar gravity
field information [3-8]. Existing high-degree lunar grav-
ity field models, such as LP100J, LP100K and LP165P,
were estimated from tracking data of LP and historical
lunar crafts before SELENE mission was implemented
[5,9,10]. It is a known fact that there are no direct ob-
servations of the farside, and the observing precision at
the Moon edge is low. Meantime the distribution of
tracking data is not even and there are more data at the
lunar equatorial area. So these lunar gravity models have
much higher resolution at the nearside and lower resolu-
tion at the farside. Also, there are more errors and alias-
ing at the truncated degrees for these models [5]. SE-
LENE can directly be tracked at the nearside and the
farside from the earth stations, and the two-/four-way
Doppler data and ranges can be collected [11,12]. A new
lunar gravity field model SGM90d to degree and order
of 90 has been developed from the SELENE-tracking
data and historical data [13]. SGM90d is the first model
including the direct observations at the farside. The lunar

J. Y. Guo et al. / Natural Science 3 (2011) 339-343 340
gravity field model is mainly used to study the lunar
air-free gravity anomalies and the Bouguer anomalies,
and discuss the lunar gravity anomalies at mascons and
basins.
The deflection of the vertical is an angle between the
direction of gravity and a reference direction [14]. Lunar
deflections of the vertical indicate the slope of selenoid
relative to the reference lunar ellipsoid, as well as the
angle between the practical lunar vertical and the normal
lunar gravity direction. So the lunar deflections of the
vertical include information of internal lunar mass dis-
tribution and anomaly. Lunar gravity vector is composed
of the vertical deflection and gravity anomaly based on a
reference normal gravity. Lunar deflection of the vertical,
as one basic observation in the lunar geophysics and
selenodesy, can provide abundant information of lunar
gravity field and selenoid. So the lunar deflections of the
vertical are very important for the study of lunar gravity
field. But we cannot directly precisely measure the ver-
tical deflections on the lunar surface at present. Lunar
deflections of the vertical and its distribution are calcu-
lated based on SGM90d in this paper. Then the correla-
tion between lunar deflections of the vertical and lunar
topography is analyzed. Differences of vertical deflec-
tions on lunar mascons and basins are also discussed.
2. LUNAR DEFLECTIONS OF THE
VERTICAL
According to the definition of Molodensky deflection
of the vertical [15], lunar deflection of the vertical can
be calculated based on the lunar gravity model from the
following formulas:


2
2
0
dsin
cossind
n
N
n
nnm
nm nm
m
GM a
r r
P
CmSm







 




(1)


2
2
0
cos
sincossin
n
N
n
n
nmnm nm
m
GM a
r
r
mCm SmP






 

 


(2)
22
VD

 (3)
tan



 (4)
where

and

the meridional and the prime vertical
components, respectively; VD the deflection of the ver-
tical; GM the lunar gravitational constant;

the nor-
mal lunar gravity;

,,r

 the spherical coordinates,
N the highest degree of the model used; n and m
degree and order of the gravity field model; nm C and
nm
S the fully normalized lunar potential coefficients;


sin
nm
P

the fully normalized associated Legendre
function, and

the azimuth of the vertical deflection.
Namiki et al. have developed a new lunar gravity field
model up to degree and order of 90 named SGM90d.
Tracking data used to determine this gravity model in-
cludes SELENE tracking data from Oct. 31, 2007 to
Apr. 1, 2008 as well as historical lunar satellites tracking
data [13]. Though has a lower degree and order,
SGM90d reveals lunar far side ring shaped gravity fea-
tures in a much higher resolution than existing lunar
gravity models like LP100k. This significant improve-
ment owes to SELENE 4way Doppler observations,
which enables direct tracking of the satellite over far
side of the Moon. The deflection of vertical and its me-
ridional and prime vertical components and direction are
calculated from SGM90d, drew with GMT [16], shown
in Figure 1.
The global distribution of meridional components of
lunar vertical deflections has been shown in Figure 1(a).
Impacted basins locating at the near side including Im-
brium, Serentitatis, Crisium and Humorum have similar
symmetrical patterns with positive meridional compo-
nents at east and negative ones at west. Nectaris and
Smythii have rather complicated distribution patterns,
which are positive-negative-positive-negative (from west
to east). Orientale, Hertzsprung, Freundlich-Sharonov,
Moscoviense, Apollo and Mendel-Rydberg locating at
the farside, on the other hand, have negative-positive-ne-
gative-positive (from west to east) distribution patterns.
The global distribution of prime vertical components
presented in Figure 1(b) also shows similar symmetrical
pattern, but with a different symmetry axis direction.
Four impacted basins, Imbrium, Serenitatis, Crisium and
Nectaris, have positive prime vertical components at
northern part of the basins and negative ones on the oth-
er haft. Humorum, Orientale and Mendel-Rydberg have
negative-positive-negative-positive distribution pat- terns
in north-south direction.
From the global distribution of lunar deflections of the
vertical (shown in Figure 1(c)), ring shaped distribution
patterns can be fond at impacted basins including Im-
brium, Serenitatis, Crisium, Nectaris and Humorum.
Vertical deflections are very large at the periphery of the
basins and much smaller in the center area. The circles
formed by large deflections of the vertical closely related
to the lunar terrain and clearly indicate the boundaries of
original mascons [13]. The large deflections of the ver-
tical at Orientale formed two concentric circles. The
outer ring is by the edge of the basin and inner one
seems to be the margin of gravity anomaly in the basin.
And it is the same case for other basins like Hertzsprung,

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-
coviense. Lorentz and Korolev have same air-free grav-
ity anomalies and Bouguer anomalies in the whole basin
area. But for Orientale, Hertzsprung, Apollo, Freundlich-
Sharonov and Moscoviense, the air-free gravity anoma-
lies are 20% - 60% less than the Bouguer anomaly [13].
Presented in Figure 1(d) is the global distribution of
azimuth of the vertical deflection. It can be seen that the
azimuth changed counterclockwise from 0˚ - 360˚ at
several famous basins with single ring shaped vertical
deflections. In the case of basins with double-ring verti-
cal deflections, each of the two ring has same changes
described above, but there’s 180˚ difference between the
outer and inner rings.
3. GRADIENTS OF LUNAR VERTICAL
DEFLECTIONS
The gradient of lunar vertical deflection is defined as
22
12
s
ss (5)
where 1
s
and 2
s
are the numerical differentials of
meridionla component and prime vertical component of
vertical deflection to latitude and longitude, respectively.
So the gradient of lunar vertical deflection can be calcu-
lated with SGM90d, shown in Figure 2. We can find that
Imbrium, Serenitatis, Crisium, Smythii, Nectaris and
Humorum have gradients with the annulus distributions.
The gradient distributions in other basins are very com-
plicated.
4. CORRELATIONS BETWEEN LUNAR
DEFLECTIONS OF THE VERTICAL
AND TERRAIN
Lunar deflections of the vertical reflect abundant in-
formation of mass anomaly and its distribution, and
closely correlate with the lunar terrain. With laser rang-
ing data (Now. 27 2007 to Jan. 22 2008) from Chang’E,
Ping et al. developed a lunar terrain model CLTM-s01 to
degree and order 360 in the form of spherical expansion
[17]. CLTM-s01 has better spatial coverage, precision
(31 m) and resolution (0.25˚) than any previous models.
Lunar terrain and its gradient derived greater than 0.5,
from CLTM-s01 have been shown in Figures 3 and 4,
respectively.
The following formula is used to calculate correla-
tions between the vertical deflections and the terrain
gradients at the nearside and farside,

J. Y. Guo et al. / Natural Science 3 (2011) 339-343 342
Figure 2. Gradients of luanr veritical deflections, the X-coordi-
nate represents longitude and the Y-coordinate represents lati-
tude.


22
DOVDOVT T
Correlation
DOVDOVT T





(6)
where DOV and T are vertical deflections and elevations
on the nearside or the farside of the Moon, respectively;
DOV and T are the mean value of vertical deflections
and the mean value of elevations on the nearside or the
farside of the Moon, respectively.
Correlations between the vertical deflections and the
terrain gradients at the nearside and the farside are 0.427,
and 0.416 respectively. From Figures 1(c) and 4, we can
find that there are high correlations between the lunar
terrain and the vertical deflections, which indicate that
the vertical deflection can be caused from the mass
anomaly in the lunar shell and surface to a large extent,
and the internal mass in the Moon may be uniformly
present. So the evolution in the internal Moon on the
whole may end.
Lunar deflections of the vertical are calculated from
SGM90d to degree and order 90, but CLTM-s01 is to
degree and order 360. Figure 5 shows the correlation
between the geopotential coefficients and the terrain
spherical harmonic coefficients to degree and order 90.
We can find that the correlations for degrees 30 to 80 are
which indicate that the lunar terrain on this scale may
largely contribute to the lunar gravity anomalies.
5. CONCLUSIONS
The lunar deflections of the vertical and their gradi-
ents are calculated from lunar gravity model SGM90d.
Analyzing different distribution patterns of vertical de-
flections at several main basins shows that basins locate
at the nearside of the Moon have different characteristics
compared with those at the farside. The differences can
be cause by the lithospheric compensation and the lunar
Figure 3. Lunar terrian derived from CLTM-s01, the X-coordi-
nate represents longitude and the Y-coordinate represents lati-
tude.
Figure 4. Gradients of lunar terrain, the X-coordinate repre-
sents longitude and the Y-coordinate represents latitude.
Figure 5. Correlation of SGM90d and CLTM-s01 to degree
and order of 90.
mantle uplifting. Relatively large basins more likely ap-
pear at the nearside of the Moon than the farside may be
caused by the inconsistency of lunar basalt activity ra-
ther than the impact [18].

J. Y. Guo et al. / Natural Science 3 (2011) 339-343 343
6. ACKNOWLEDGEMENTS
This study is supported in part by the National Natural Science
Foundation of China under grant No. 40974016 and 40974004, the
Key Laboratory of Surveying and Mapping Technology on Island and
Reef of SBSM, China under grant No. 2009A02, the Research & In-
novation Team Support Program of SDUST, China, and the Science &
Technology Development Program of SBSM, China.
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