Chapter Two Normal and Anomalous Gravitational Fields of the Moon

Chapter Two

Normal and Anomalous Gravitational Fields of the M o o n

2.1 Structure of the Gravitational Field and Its Role in the Evolution of the Moon

The manner in which the Moon’s gravitational field is distributed is rather complex. It reflects the irregularities of the internal structure of the Moon and its figure. However, if the lunar gravitational field is examined in its entirety, one of its striking features is the central symmetry with respect to the Moon’s centre of mass. This symmetry is almost perfect. Indeed, variations in the gravity potential on the lunar surface do not exceed 0.02% of its mean value W = 282 x 1O’O cm2 s-’, while those in the acceleration of gravity do not exceed 0.3% of its mean value equal to 162 cm s - ~ .How could Nature create an entirety so perfect in the midst of the cataclysmic processing that has given rise to the Solar system, including the Moon? Such an ideal gravitational field must have been the result of eons of the Moon’s evolution. Doubtless, the lower absolute value of lunar gravity, as compared to that of the Earth, has made this parameter secondary to other forces determined by temperature, pressure and other factors. This may be the reason why the Moon is farther than the Earth from the state of hydrostatic equilibrium and isostatic compensation: its body retains higher stresses, and the anomaly of its gravitational field is more pronounced. The fact that lunar gravity is lower than terrestrial gravity is also responsible for many other processes and phenomena. Gravity is known to be the major factor determining the presence of an atmosphere on planets because it prevents gas molecules from being scattered. For these molecules to be permanently retained near the

100

Lunar Gravimetry

planet surface requires that the parabolic velocity

v = (2gR)”2 where g is gravity and R is the planet radius, should be far in excess of the root-mean-surface velocity of the gas molecules. Since the lunar gravity g is low, the parabolic velocity is also low: V = 2.38 km s-’. This has resulted in the Moon having lost its atmosphere. The low relative value of g has also been partially responsible for the jagged contours of the Moon’s relief. The lunar gravity cannot overcome the molecular cohesion of the lunar rock matter, whereas on the Earth, where gravity is six times as high as on the Moon, such rock formations would tend to collapse. The Same applies to the steeper slopes of aggregations of loose materials on the Moon, as opposed to the Earth. Another result of the low lunar gravity, albeit indirect, is the peculiar Moon’s landscape. The already mentioned loss of the atmosphere has brought erosion processes to a halt with the consequence that the surface features have remained intact. On the other hand, the lack of atmosphere has allowed meteoric bodies to impinge upon the lunar surface unhindered, and their impacts have pitted the Moon. Examination of the gravitational field over the spherical surface of the Moon makes it evident that its irregularities (anomalies) are superimposed on a permanent gravitational field of a sufficiently high level. It can be readily seen that the anomalies vary in extent. The Moon has long been approximated by a triaxial ellipsoid, which is borne out by observations. The semiaxes of the ellipsoidal Moon differ in length by no more than a kilometre. This difference is responsible for the most extended anomalies in its gravitational field. The next largest anomalies are due to maria and highlands, and these are followed by anomalies due to mascons and other tectonic formations of comparable size. No matter how irregular the variations in the anomalous portion of the gravitational field are, certain statistical characteristics and the pattern of these variations can be established. To solve the various problems associated with these aspects, it is convenient to represent the lunar gravitational field as consisting of two parts-normal and anomalous. The former corresponds to a simple lunar model closely approximating the real Moon. It represents the figure, internal structure, and gravitational field of the Moon without any detail, in a most general form. The changes in the normal field follow a simple pattern which depends on the observation point coordinates and a few constants. As can be inferred from the foregoing, the anomalous gravitational field is irregular in structure. The anomalous field can be regarded as the difference between the actual gravitational field and what has been adopted as the normal one. The division of the gravitational field into normal and anomalous parts is a

101

Normal and Anomalous GravitationalFields

convention and depends on the principle underlying the definition of the normal field. The ratio between the anomalous and normal parts of the gravitational field is to a great degree dependent on the field characteristics under consideration (gravity potential, gravity itself, gravity gradients, third potential derivatives,etc.). The normal part of the gravity potential is several thousands of times greater than the corresponding anomalous part, while the normal vertical gradient is only several tens of times that of the anomalous gradient. As regards the third derivatives of the lunar gravity potential, their normal part is substantially smaller than their anomalies. Also different is the spectral composition of the various derivatives of the lunar gravity potential. In this chapter, we shall analyse the expressions for these derivatives, which are essentially expansions in terms of spherical functions. At present, the gravitational field of the Moon is known in detail given by spherical functions to within the 16th order, except for the narrow areas defined through observations via Apollos 14, 15, 16,17, and the Apollo 15 and 16 subsatellites from altitudes of several tens of kilometres above the Moon’s surface. 2.2 Expansion of Lunar Gravity Potential Derivatives in Spherical Functions

As the initial expansion, let us use an expression for the lunar gravity potential, including the potential of lunar mass attraction and the constant portion of the centrifugal-tidal potential due to the Moon’s rotation and the gravitational tidal effect of the Earth

x (Cnm cos m l

+ S,,

sin ml)P,,(sin cp)

+ GAM @p 2[-1 - 2Pzo(sincp) +

I

cos 21P22(sin4711

(2.2.1)

The centrifugal-tidal potential is the term in square brackets. Let us consider the derivatives of the lunar gravity potential at points ( p , c p , l ) of its surrounding space. To determine the radial derivative, differentiate (2.2.1) with respect to p:

- - - -m=2

x

(Cnm

cos m l

+ S,,

m=O

sin ml)P,,(sin cp)

1

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Lunar Gravimetry

+GMce p [- - 3Pzo(sinq) + +Pzz(sincp) cos 2 4 A3

(2.2.2)

This derivative has the dimensions of acceleration. It is the major component of gravity acceleration. Now, let us find the other two acceleration components normal to the p axis. One of the components is tangential to the parallel:

(2.2.3) where

The other coefficients of this series are:

K!,: = mSnm,

MLZ = -mC,,

The remaining component is tangential to the meridian:

-- -pdcp

pz

m=O

n=z

+ Snmsin mL)[ -m tan cpPnm(sincp) + 6Pn,,+ I(sin cp)] GMce P (2.2.4) [+PZ1(sincp) + tan qPzz(sin cp) cos 2L] A3

x (Cnmcos rnn

Equation (2.2.4) has been derived using the well-known formula

where 1 0

atm
The complete horizontal acceleration component a W/al will be determined if we find

ai

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Normal and Anomalous Gravitational Fields

This quantity also has the dimensions of acceleration. In the case of a uniformly spherical Moon, a W p l = 0. Next, consider the higher derivatives of the potential W.The second radial derivative is given by a2W

2GM,

dp2=p3

{ +- (”)’ (;’’ - + 1 1

p-’

O0

A

3

-

n=2

It

m=O

(KfL’cosmA

+ Mf$” sin mA)P,,(sin

cp)

(2.2.5)

where

and the other coefficients are

Find the mean value of the second radial derivative on a sphere with the radius p : -

-a2w ap2

j’ja2w(s)

1 4np2

ap2

S

-

The obtained value of a2 W/ap2may be regarded as the normal second radial gradient of the gravity potential, corresponding to a spherical Moon with an even or concentric density distribution. The second term in square brackets is due to the tidal effect and equals 2.5 x The quantity dZW/ap2 also varies with the distance from the Moon’s centre. Table 2.1 lists the values of a* W/ap2,calculated without taking into account the centrifugal-tidal effect. The selenocentric constant of gravitation has been assumed to be G M , = 4902.7 km3 s - ~ . Finally, consider the third radial derivative of the lunar gravity potential

(;y

-a3w - y{1 f + aP3

n=2

m=O

x (KIP,PP) cos m l

+ MkCP)sin mA)P,,(sin

cp)

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Lunar Gravimetry

TABLE 2.1 Second and third radial gradients of the lunar attraction potential at different distances p from the Moon’s centre

- 1874.2 - 1870.9 - 1861.1 - 1225.6 -78.3 - 9.8

1736 1737 1738 2000 5000 10,Ooo

where KfW) =

M?,PP) =

(n

323.9 323.1 322.4 183.8 4.7 0.29

+ l)(n + 2)(n + 3 ) C., 6

(n

+ l)(n + 2)(n + 3) S”ln .

6

The values of the derivative a3 W/ap3= -6GM,/p4, corresponding to a uniformly spherical Moon at Merent distances from its centre are given in Table 2.1. The units of this derivative are cm-’ s-’. No instruments have been developed as yet for its direct measurement, but it can be calculated given a potential or gravitational field. As can be seen from the expressions for the coefficients K;:) and Mi:), in the case of the second radial potential derivative a’ Wlap’, they contain the factor ( n l)(n 2)/2, while the coefficients KfLP) and M f g P )in the case of a3 W/ap3contain the factor ( n + l)(n + 2)(n + 3)/6. In other words, one may expect high values of the short-wavelength components of the second, and more especially third, derivatives of the lunar gravity potential. Detailed observations of the Moon’s gravitational field distribution lend support to this assumption. The gravity anomalies shown in Fig. 1.9 correspond to horizontal gravity gradients as large as hundreds of eotvos units (E). The anomalies of the second radial derivative of the gravity potential may be of the same order of magnitude. The above expressions for the derivatives permit their values to be determined at random points in the space around the Moon. To determine them at points on a particular surface (physical surface of the Moon, selenoid, ellipsoid, etc.), one must solve these equations together with that for the radius vector p ( q , A) describing the corresponding surface. Their solution implies exclusion of the radius vector p. Then, the formulae for the derivatives will be functions of only two angular variables rp and I and will give the distribution of these derivatives on respective surfaces. ~

+

+

Normal and Anomalous Gravitational Fields

105

2.3 Expansion for Gravity

It is appropriate now to consider at greater length the derivation of a formula to define gravity g(p, cp, A). Apparently, at any point it can be determined as the square root of the sum of squares of the components along the three orthogonal coordinate axes p, cp and k,

(2.3.1) Since (a W/ap)’ is substantially greater than the other two addends in the right-hand side of (2.3.1), the equation can be rewritten as

This expression represents the first two terms of the expansion of the righthand side of (2.3.1) in terms of the degrees of the relation of the above addends to the square of the radial derivative aW/ap. Using (2.2.3) and (2.2.4), we shall write the expressions for the squares of the components (l/p)(a W/acp),(l/p cos cp)(a WpA). Only those terms will be retained in the namely, the terms with expansions whose relative value exceeds 1 x coefficients CiO, c:2,c20c22 and CZOc31

(2.3.3)

The expansion of gravity into a series will be performed as Bursa (1975) did for terrestrial gravity. Our expansion is different in that we retain the terms (2.3.3) essential for the Moon. Thus, confining ourselves in the expansions for the squares of (l/p)(d W/ap)and (l/p cos cp)(a W@A)only to the terms (2.3.3),

1

106

Lunar Gravimetry

Here and in what follows, we omit the argument sin cp of the Legendre functions P,,(sin cp) for brevity. Substitution of (2.3.4) into (2.3.2) and simple transformation give

In order to eliminate the squares and products of the associated Legendre functions, we shall use the following auxiliary relations:

3 72 6 6 tan2 cpp:, = - - P~~+ 3P22 = -- p40+ - pzo+ 35 35 7 5

4 7 tan cpPZ1 P31 = - PS1 -P31 7 5

+

1 cos2 21 = 2

+

36 -P l l 35

+ cos4n 2

1 cos4n sin2 2A = - - 2 2

These relations enable (2.3.5) to be transformed into g(p, cp,

n) =

1

aaPw {

~

+

(X -

- (c:,

+12c9

(2.3.6)

107

Normal and Anomalous Gravitational Fields

6 12 c20c22~22 cos 2a - - c20c22~42 COS 2a 7 35

--

Substitute expression (2.2.2) for a W/ap into (2.3.7). The transformations give the final expression for the distribution of gravity g at random points (p, cp, A) on the Moon's surface:

The other harmonic coefficients take the form gnrn

= (n

+ l)Cnrn,

hnrn

= (n

+ 1)S.m

(2.3.10)

Thus, the coefficients(2.3.9) and (2.3.10) of expansion of the lunar gravity g are expressed in terms of harmonic coefficients C., and S.,.

108

Lunar Gravimetry

2.4 Selenoid

First of all, a definition must be given to the figure of the Moon. Distinction is made between several figures, each representing particular properties of the Moon, including the geometrical, dynamic, hydrostatic, equal gravity potential (equipotential), equal gravity, and other figures. By the geometrical figure the physical surface of the Moon is understood. All other figures are defined by some imagined surfaces. The dynamic figure is an ellipsoid exhibiting certain dynamic properties displayed by the moving Moon. The hydrostatic figure of the Moon is characterized by an equal-pressure surface. Also used is a figure with equal values of the gravity potential at all points on its surface, known as the level surface. Of great importance in lunar studies may be surfaces of equal gravity, equal radial gravity gradient, and so on. Depending on the problem to be solved, each figure is examined in every detail or a smoothed surface is considered, only generally representing the true figure but simple in shape (sphere, ellipsoid, spheroid) and, therefore, convenient to use. The dynamic figure is always an ellipsoid, namely, that of the inertia of the Moon. All of the above-mentioned figures of the Moon will be discussed below. Here, we should like to dwell on the level surface whose equation is derived by equating the expression (2.2.1) for gravity potential to a constant Wo and the resulting expression is regarded as an equation for a closed surface. This will be an equation with three variables p, @ and R. Varying Wogives other level surfaces. In the case of the Earth, one of the level surfaces coinciding with the averaged surface of the seas and oceans is known as the geoid. By analogy, the Moon is represented by the selenoid. The absence of a water surface on the Moon does not complicate the solution of the selenoid definition problem. In so far as the selenoid is an arbitrary surface, it may be assumed to represent one of the level surfaces sufficiently close to the averaged physical surface of the Moon. In order to fix it, a level surface passing through a point on the Moon is considered. For example, the selenoid may be tied to a point where gravity is known from direct measurements. If G M , is a known value and go is the measured gravity on the Moon’s surface, the constant Wo may be found from

w, =

~

RO

=

(GM,go)’/’

Thus defined, Ro may be named a dimension factor (Bursa, 1969). The level surface equation takes the form

+ S,,

sin m,l)P,,(sin cp)

109

Normal and Anomalous Gravitational Fields

Pzo(sincp)

+ 21 cos 2;1Pz2(sincp)

= GM/Ro

-

(2.4.1)

To describe a surface in terms of (2.4.1) is not convenient, which is why it has to be transformed so that the radius of the surface is a function of the angular coordinates cp and 1,that is, P =

4

It is difficult to find an exact solution of (2.4.1) with respect to p because we are dealing with l/p to various powers. Here, the method of successive approximations is used. Retaining the terms with the squares and products of (2.3.3) in the expansion gives p(cp, A) = R O { 1 +

z2(g)’

m=O

(c,, cos rnA

Pzo(sin cp)

+ s,, sin rnl)P,,(sin

+’-21 cos 21Pz2(sincp)

+ 2C20C22cos 21PZo(sincp)PZ2(sincp) + C:,

I

cp)

1

cos’ 2IP:,(sin cp)] (2.4.2)

C2oC31 cos lPzo(sin~ ) P ~ ~ (cp)s i n Using the auxiliary relations (2.3.6) as well as the relations 18 Pl0 = jjP40

+ -27P z o + -51

3 72 48 Pg2 = 35P44= - P40 - -P20 35 7 3 2 PZOPZZ = -p42 - - p22 35 7

+ 245

-

(2.4.3)

and grouping the terms with the same orders of spherical functions, we can write with the following expression for the radius vector p of the level surface:

110

Lunar Gravimetry

where the coefficients A,, and B,, are related to the harmonic coefficients C,,, and S,,, as follows: 24

All

= --(-)CZOc31 18 R

7 Ro

A44

=

(&-[c4.

A 5 1 =(&-[c51

-7cZOC3l lo

1

For the rest of the coefficients included in expansion (2.4.4), the following expressions hold: (2.4.6) Thus, expression (2.4.4) respresents the radius vectors of points on the selenoid surface as an expansion in spherical functions. The coefficients of this expansion depend on the harmonic coefficients C,, and S,, of the Moon's gravitational potential, on the value of p - l which is the ratio of the Earth's mass to that of the Moon, and on the adopted constant Wo of the gravity potential. The following features of expansion (2.4.4) should be noted. The mean radius of the selenoid becomes equal, as can be seen from expansion (2.4.4), to

Normal and Anomalous Gravitational Fields

111

R* = RoAoo instead of Ro. Assuming that R/Ro = 1, p-’ = 81.30, Ro = 1738 km, and A = 384,400 km and using formula (2.4.5), we shall find the mean radius increment AR = R* - Ro = 435 cm. The term with the coefficient A l l , which appears in expansion (2.4.4), is indicative of the selenoid volume centre shifting along the x axis with respect to the initial origin of coordinates, coinciding with the Moon’s centre of mass. The amount of this shift has been found to be about 1 cm-negligibly small. The nonlinear corrections to the harmonic coefficients C,, and S,, in the coefficients A,, and B,, (2.4.5) are small, too. The above formulae have been derived for estimations and practical use in the future when more accurate measurement data become available. Since the selenoid is a three-dimensional figure, it is represented as a map of elevations of its surface above the reference surface of a sphere, ellipsoid, or spheroid. Figure 2.1 shows the isolines of selenoid elevations above the sphere with R = 1738 km for the near and far sides, derived from the results presented in $1.13 of Chapter 1. It can be seen that the elevation of the selenoid surface varies within f500m. It should be remembered for comparison that the variation in the elevations of the geoid over a triaxial ellipsoid (Fig. 2.2) does not exceed & 100 m. The geoid is a more regular figure than the selenoid. This becomes more apparent if the variations in the elevations of the geoid and selenoid are related to their radii. In terrestrial measurements, the geoid is a necessity because the elevations of the Earth’s surface are obtained through levelling with respect to the geoid (sea level). The geoid is required as an intermediate surface used as a reference in measuring elevation of the Earth’s surface. Those of the Moon’s surface are determined differently. They are measured by an externally stationed observer virtually as geometric distances between points on the physical surface of the Moon and a geometric centre. Therefore, there is no need to use the selenoid in studying the geometrical figure of the Moon. 2.5 Figure of the Normal Moon

The solution of most problems can be simplified if a simpler level surface is involved instead of the rather complicated selenoid. Corresponding to this surface will be a simpler external gravitational field representing the real field minus small-scale details. This simple surface may be a sphere, a spheroid, an ellipsoid of rotation, a triaxial ellipsoid or whatever. In this case, the angular velocity of the model of the Moon is assumed to coincide with that of the real Moon. It is also assumed that their masses and the parameters representing the invariable part of the tides due to the Earth’s gravitational effects are equal as well. The model of the Moon satisfying all these conditions is

Fig. 2.1. Map of the moon showing selenoid elevations relative to the surface of a sphere with radius R = 1738 km (isolines taken at 100 m intervals). Scale 1 : 20,000,000.

Fig. 2.2. Geoid elevations relative to an ellipsoid of rotation with a polar flattening of 1/298,256 (isolines taken at 10 m intervals)

114

Lunar Gravimetry

referred to as the normal Moon, and its gravitational field is known as the normal gravitational field. That part of the gravitational field which is governed by the irregularities in the Moon’s internal structure and figure is called the anomalous gravitational field. It is smaller in magnitude and has an uneven, irregular distribution. Two approaches are used in constructing the figure of the normal Moon and its normal field. The first approach involves expansion of the real lunar gravity potential in spherical functions (2.3.l), in which several principal terms are retained. The resulting gravitational field is regarded as normal. If the truncated expression for the gravity potential is equated to a constant, the result will be an equation for the surface of the normal Moon’s figure. We have to decide which terms of expansion (2.3.1) can be considered principal. This is a problem because it is difficult to identify one or two dominating terms in expansion (2.3.1) of the lunar gravity potential. If too many terms are assumed to be principal, the normal Moon’s figure will become too complicated. As is known from observations, the most pronounced gravity anomalies on the Moon are given by harmonics with Pzo(sin cp), cos 2RPz2(sincp), and cos RJ’31(sin’cp). Therefore, taken as the surface of the normal Moon may be-a level surface given by a shortened potential expansion:

C31 cosAP,,(sincp)

x

[--2 3

5 Pzo(sin cp) 3

+ -21 cos 2APz2(sincp)

(2.5.1)

Apart from the terms associated with the lunar mass attraction, retained in this expression are those representing the invariable part of the centrifugaltidal potential. To find the normal Moon figure, equate the right-hand side of (2.5.1) to a constant GM,/R,. The resulting expression for the radius of the normal Moon’s surface is a particular case of formula (2.4.2) at n < 6. The coefficients of expansion in spherical functions are given in formulae (2.4.5). Since formula (2.5.1) giving the gravity potential of the normal Moon coefficients S,,, only coefficients A,, are left in the equation of the normal Moon’s gravity potential. The expression for the radius of this surface will are taken into have the following form if terms with magnitudes up to account:

115

Normal and Anomalous Gravitational Fields

+ [A1lPll(sin cp) + A31P31(sincp) + A5,PSl(sin cp)] + CA22P22(sin cp) + A42P42(sincp)] cos 2A + A44P44(sincp) cos 4A}

cos A

(2.5.2)

The equation contains zonal, tesseral and sectorial spherical functions; that is, more than the simple “triaxiality” of the normal Moon is taken into account. The polar flattening is given by the terms with A 2 , and A40. Of paramount importance is the harmonic with A2,, which is greater by almost two orders of magnitude than that with ,440. The purely longitudinal variations in the radius of the surface are given by the terms with A 2 2 and A44. In addition to these two harmonics, the axial asymmetry of the surface is provided by those with A 3 1 , A41 and AS1; that is, harmonics whose second subscripts are nonzero. The term with A J 1 is predominant. The other approach to constructing the normal Moon’s figure resides in the following: a figure, such as an ellipsoid of rotation or a triaxial ellipsoid, is chosen first. The ellipsoid is assumed to be a level (equipotential) one, which means that the gravity potential on its surface is constant. It is also assumed that the mass, angular velocity of rotation and tidal effect of the Earth are equal for the real Moon and the model. The parameters of the ellipsoid representing the normal Moon are determined from the following condition:

(2.5.3) where p and pe are the radii of, respectively, the selenoid and normal ellipsoid surfaces. The ellipsoid of rotation is characterized by two parameters: semimajor axis a and polar flattening c1 = (a - c)/a; four parameters determine the figure of the triaxial ellipsoid semimajor axis a, polar flattening a, equatorial flattening a1 = (a - b)/a, and longitude A, of the semimajor axis. The equation of the triaxial ellipsoid in a rectangular system of coordinates is

-+ u2 X2

Y2

Z2

aZ(1 - a1)2 + a2(1 - a)2 =

(2.5.4)

The coordinates x, y and z can be expressed in terms of the polar coordinates p, cp and A: x = pe cos cp cos (A - A,) y = pe cos cp sin (A - A,) z = pe sin cp

116

Lunar Gravimetry

These coordinates are substituted into (2.5.4), which is expanded into a power series in terms of a and al. Retaining the terms of the second order of smallness with respect to a’, a: and aal gives pe(cp, A) = a(1 - a ) ( l - a1)[ 1

+ +E(2 - a ) cos2 cp

+ +a1@ - a l ) sin’ cp + f a l ( 2 - al)sin2 cp

+ )a1(2 - a1)(1 - a)2 cos2 cp cos2 (A - 10) + $a2 C O S ~cp + $a: sin4 cp + $a; C O S ~cp C O S ~(A - A,) + 3aal sin’ cp cos2 cp + 3aa1 C O S ~cp cos’ (1 - no) + 3 4 sin2 cp cos’ cp cos2 (A do) -

(2.5.5)

The following relations

+ cos 2(A - A,)]/2 sin’ cp = (1 + 2PzO)/3 (A - no) = f + 3cos (A - no) + + C O S ~(A - A,)

cos2(A - 2,) = [l

COS~

(2.5.6) c0s4

cp = 8 15 - ‘2 16 p20

+AP4O

= 3p22 - &p42

sin’ cp cos’ cp = &

+ &pz0

=h

p44

- & ~ 4 0=

+ &P42

hpZ2

are used to reduce (2.5.5), giving the radius of the triaxial ellipsoid surface, to the following form:

+ azoPzo + (az2cos 21 + bZ2sin 2 4 P Z 2 + a40P40 + cos 2A + bzz sin 21)PZ2 + cos 22 + b42 sin 2A)P42 + (a44cos 4A + b44 sin 4A)P44

pe(cp, 1) = a00

(a42

(a42

where aoo = a[1 a20 a40

-

3.

+ = a[%a2 + = a[-ja

-

ial - $2

3.1

- 3.2

- l5m 21 - 5ma11 4

+ &a: + 3aaJ

- +&la]

(2.5.7)

,

(2.5.8)

Normal and Anomalous

117

Gravitational Fields

The equation representing the difference between the radii of the selenoid (2.4.5) and the normal Moon’s ellipsoid (2.5.7) is written as follows:

dcp,4- P ~ ( P4 , = Ro* - aoo + R,*All cos AP21(sinrp) N

n

1 C(Ro*Anm - a n m ) cos mA + n1 =2 m=O

+ (R,*Bnm- brim) sin mA]P,,(sin

(2.5.9)

cp)

The radius pe(cp, A) (2.5.7) of the triaxial ellipsoid is related, through coefficients anmand b,,, to several parameters, namely; semimajor axis a, polar flattening a, equator flattening a, and longitude loof the semimajor axis. By varying these parameters one can obtain a triaxial ellipsoid approximating the selenoid as closely as possible. As is known from the previous section, the selenoid is derived from the observed gravitational field of the Moon. Therefore, the coefficients R;Anm and R,*B,, in (2.5.9), coresponding to the selenoid, are considered to be known. The unknowns of interest are the four parameters of the triaxial ellipsoid: a, a, a, and Lo. When the parameters of an ellipsoid of revolution are determined, it is assumed that a, = l o= 0. When a spherical surface is selected as that of the normal Moon, only one parameter needs to be found radius R = a. In his work, Buzuk TABLE 2.2 Maximum values of some characteristicsof the lunar gravitational field, calculated relative to different

reference surfaces

Gravity anomalies,

Perturbing potential,

Ag (mGal)

T (lo6 cm2 s - ~ )

Relative to a sphere -240-+135 - 10,796-+ 8665 Relative to an ellipsoid of rotation -24O-+ 163 -9755-+6438 Relative to a triaxial ellipsoid - 10,734-+ 5383 -246-+ 137

Plumb-line deflection components

t (“1

v(“)

+221

- 1 8 6 + 263

-213-+ 188

- 186-+ 263

-259-

-201-

+200

-235-+251

118

Lunar Gravimetry

(1975) selected such surfaces which satisfy the minimum condition (2.5.3). The selenopotential model proposed by Gapcynski et al. (1969) was used in constructing the normal Moon’s figure. As a result, the parameters of the optimal sphere (R = 1736.41 km), optimal ellipsoid of revolution (semimajor axis a = 1736.68 km and polar flattening a = 1/3147), as well as optimal triaxial ellipsoid (the same semimajor axis a and polar flattening a, al = 1/5028, and lo = 3” W) have been determined. Table 2.2 lists the maximum values of the perturbing potential T, gravity anomalies Ag, and plumb-line deflection components 5 and q, corresponding to these three surfaces. It can be seen it does not actually matter which surface has been chosen to represent that of the normal Moon. 2.6 Distribution of Normal Gravity

In accordance with the two drastically different approaches to constructing the surface of the normal Moon, the field of normal gravity y can be formulated in two ways. In the case of the first approach, gravity is determined from formulae similar to (2.3.1) and (2.3.8). The only difference is that a “shortened” potential W is used instead of the initial gravity potential (2.2.1). This will be a gravity potential including, in addition to the principal zeroth harmonic of G M , / p , those harmonics with Pzo(sin cp), Pz2(sincp) cos 22, Pz2(sincp) sin 2A, PJl(sin cp) cos A and PJl(sin cp) sin A. The acceleration due to normal gravity y will be determined similarly to gravity g(p, cp, A) in $2.3, with the difference that only the harmonics mentioned above will be retained:

+ hz2 sin 2A)PZ2+

1

+ hJ1 sin A)PJ1

(3

(931

cos A (2.6.1)

The coefficients go, g20, g22, h z 2 , 9 3 1 and hJ1 are expressed in terms of equations (2.3.9). Let the normal Moon’s surface be that of a spheroid:

Acp, 4= ROC1 + CzoPzo(sin cp)l

(2.6.2)

This equation can be used to eliminate p from (2.6.1), bearing in mind that coefficients (2.3.9) are also dependent on p. Performing the ncessary transformations and assuming that R = Ro, we obtain the following

119

Normal and Anomalous Gravitational Fields

expression for normal gravity at points on the normal Moon's surface (2.6.2):

where

(2.6.4)

y g = -35 162 c:, yk:' =

+ 76 P - 1 c 2 0 ( 3

3

-3s 36 c 2 0 c 2 2 - 70 3 p- 1 c 2 , ( 33 40

yyi = --7 c2 0 c 3 1 Normal gravity y at points on the normal Moon's surface (2.6.2) becomes dependent only on latitude cp and longitude 1.The predominant role in the distribution of gravity is played by the harmonics included in the initial equation (2.5.1). The additional harmonics P l l cos 1, P42cos 21 and PS1cos 1,as well as the small corrections to the coefficients of the principal harmonics, have emerged as a result of retention of the terms (2.3.3) in the expansions. The resulting variation in gravity does not exceed 0.01 mGal. The dimension factor Ro calculated from the measured values of lunar gravity (see Table 1.8) is not representative of the Moon in its entirety. The

120

Lunar Gravimelry TABLE 2.3 Values of the coefficients in the normal gravity formula

Values of the codlicients in

Y E:,:Y

Ferrari's model (1977)

The generalized model 0.515 x -0.2044 x 0.683 x -0.447 x 0.980 x 0.218 x 0.193 x 0.479 x 0.286 x

0.615 x -0.1923 x 0.621 x 0.756 x 0.1168 x 0.259 x 0.192 x 0.457 x 0.341 x

10-3 loW4 10-5 10-4

lo-'

1O-j

lo-' 1O-j 10-4

value of Ro is underestimated because gravity was measured only on the Moon's near-side surface, which is nearer to its centre of mass than the farside surface. In our calculations of the normal gravitational field, the dimension factor Ro in formulae (2.6.4)was assumed equal to 1738.0 km, the mean radius of physical surface of the Moon. Using the harmonic coefficients C,, and S,, listed in Table 1.13, taking the specified value of R o , and assuming that Ro/A = 4.57 x lop3and p-' = 81.3, we have calculated the coefficients yb',' and:;7 (2.6.4), and the results are given in Table 2.3. Table 2.4 lists the values of normal gravity y ( q , A) for some points on the surface of a symmetrical normal Moon having the shape of a spheroid. Long before the lunar gravitational field began to be explored from satellites, Grushinsky and Sagitov (1962) derived the formula for normal TABLE 2.4 Values of the normal gravity y ( q , A) at various points on the normal Moon's surface, in mGal

rp (deg)

0

60

120

180

240

300

-90 -60 -30 0 30 60 90

162,285 162,330 162,327 162,310 162,327 162,330 162,284

162,302 162,308 162,300 162,308 162,302

162,271 162,304 162,325 162,304 162,271

162,263 162,316 162,358 162,316 162,263

162,257 162,301 162,333 162,301 162,257

162,291 162,307 162,311 162,307 162,291

The coefficients from the generalized model (Table 1.13) have been used

121

Normal and Anomalous Gravitational Fields

gravity distribution: y(cp,

1) = y.(1

- 0.000,37 sin2 cp

+ O.O00,08 cos2 cp cos 2 4

(2.6.5)

They used the differences in lunar inertia moments, known from astronomical observations of the Moon. The latter formula has led to the interesting finding that lunar gravity diminishes from the equator poleward in spite of the Moon’s polar flattening, while in the case of the Earth gravity increases from the equator toward the poles. The decrease in the normal lunar gravity is about 60 mGal, the range of its longitudinal variation along the equator being about 25 mGal. If only terms with yi‘ and yil.J are left in the normal gravity formula (2.6.3) and the numerical values of harmonic coefficients C,, and S,, from Table 1.13 are used, then y(cp,

1) = 162,306(1 - 0.000,92 sin2 cp

+ 0.000,077 cos2 cp cos 21) mGal (2.6.6)

In deriving expression (2.6.6), it was assumed that in formulae (2.6.4) Ro = 1738.0 km, A = 384,400 km, G M , = 4902.71 x lo9 m3 s - and ~ p - ’ = 81.30. So the poleward decrease in gravity, which looked paradoxical in 1962, has proved to be a fact. The other approach to constructing a normal gravitational field is based on the derivation of formulae descriptive of its distribution over a body of a preselected shape, which is usually an ellipsoid of revolution or a triaxial ellipsoid. It is assumed that we are dealing with a level ellipsoid, which is to say that the gravity potential is constant all over its surface. The formula for gravity distributions over the surface of a triaxial level ellipsoid was derived for the first time by Mineo: ~ ( c p ’1 ‘

+ by, cos2 cp sin2 1 + cy, sin2 cp cos2 cp cos2 1 + b2 cos2 cp sin2 1 + c2 sin2 cp)1/2

ay, cos2 cp cos2 i (a2

(2.6.7)

where a, b and c are the semiaxes of the ellipsoid, y,, Y b and yc are the normal gravities at the ends of the semiaxes. The known Somigliana’s formula for a level ellipsoid of revolution is a particular case of Mineo’s formula. Using the smallness of the following quantities: u = (a - c)/a (polar flattening of the ellipsoid), u1 = (a - b)/u (equatorial flattening), B = (y, - y,)/ya and p1 = (y, - Yb)/Ya (ratios of the gravity differences at the ends of the semiaxes to gravity y,), one can express (2.6.7) as a power series of these small values. In the expansion for the Moon, the terms with a l and PI must be retained to the same order of smallness as tl and p. Restricting ourselves to the second order of smallness of u, /I, u1 and B1, we obtain the following formula for the

122

Lunar Gravimetry

distribution of normal gravity y over the surface of a triaxial level ellipsoid: ~ ( c p , 1.) = 70Cl + ~20P20(sin cp) + y40(sin cp) + 722 P d s i n cp) cos A

+

742 P42(sin cp)

cos 21 + y44P44(sincp) cos 4A]

(2.6.8)

where 70 =

7eCl

+ 9 + +Pl + &-2@

- 2UlP1

+ aal + a l p + pla)] = [@ - fPl + &(a: - a2 2aB - alpl + Mal + a l p + p l a ) ] 3 a2 3 1 alp1 + + - ci: - - (ala + a l p + p l a )] 35 2 16 2 - a2 - a:

720

-

-

g-sl + h a : + $(%PI aal - .1P 1 1 2 742 = i K d T a 1 + alp1 aal - up1 - alp) 1 1 2 744 = m ( r a + N l P d Y22 =

-

(2.6.9)

-P141

-

The similar relations derived by Zhongolovich (1952) for the Earth are a particular case of (2.6.9). To distinguish between the two approaches just described, the former can be defined as “Helmert’s” and the other as “Clauraut’s”. It is by resorting to the first approach that the famous German geodesist Helmert (1 843-1917) derived, from the scant terrestrial gravity measurements at that time, the first formula of normal Earth gravity distribution which still enjoys wide currency. The theory of construction of the external gravitational field of a planet with a preselected level surface was set forth by the remarkable French scientist Alexis Clairaut (1 7 13-1 765) in his book “Theorie de la figure de la terre tir6e des Principes de l’hydrostatique” (Theory of the Earth’s Figure, Based on Fundamentals of Hydrostatics) published in 1743, which was a landmark in the theory of planets’ figures. This theory is based on the second approach to constructing the normal gravity field of a planet.

2.7 Surfaces of Equal Gravity and Equal Radial Gravity Gradient

The same approach used in deriving the equipotential (level) surface can be used to determine the surface of equal gravity and that of equal radial gravity derivative or, in other words, vertical gravity gradient. Sagitov and Tadzhidinov (1983) developed equations for radii of surfaces of equal gravity and equal radial gravity derivatives and obtained the isolines of elevations of these surfaces. In doing so, they equated the right-hand sides of (2.3.8) and

123

Normal and Anomalous Gravitational Fields

(2.2.5) to constants. This procedure yields implicit equations for these surfaces. The equation of equal gravity surface takes the form

x

[+ 1

y:(

$l

io

(g., cos m 1

+ hnm sin mA)P,,(sin

cp)

1

= go

(2.7.1)

For the surface of equal vertical gravity gradient we have

(2.7.2) In deriving the radii of these surfaces, we shall restrict ourselves to the linear terms with respect to g,,, h,,, K,, and M,,, while ignoring the terms representing the centrifugal-tidal . effect. Equation (2.7.1) will give us an explicit equation of the surface, in which the radius pe of the equal gravity surface is expressed in terms of the angular coordinate cp and 1:

+ h,,

sin mA)P,,(sin cp)

1

(2.7.3)

1

(2.7.4)

where po = (GM/go)1/2. The coefficients grim and h,, are expressed in terms of (2.3.9) and (2.3.10). Equation (2.7.2) was used to find the radius pe, of the surface of constant radial gravity gradient:

+ M,,

sin ml)P,,(sin cp)

for ghl) = 162,718 mGal the equal gravity surface is represented in Fig. 2.3a in the form of elevation isolines with respect to a sphere having radius pb1) = 1738.0 km. The spread of elevations of the equal gravity surface is more pronounced than that of elevations of the selenoid. In this case, the elevations are as high as 1500 m. Yet more pronounced must be the spread of elevations of the surface of equal radial gravity gradient (Fig. 2.4a). Let us now see what form a surface on which gravity equals half that

124

Lunar Gravimetry

-120"

-60"

0"

60"

120"

180"

(a) Fig. 2.3. lsolines of equal gravity surface elevations: (a) g = 162,718 mGal, the elevations are taken relative to a sphere with pi" = 1738 km (isolines taken at 500 m intervals); (b) g = 81,356 mGal, the elevations are taken relative to a sphere with p t ) = 2454.82 km (isolines taken at 100 m intervals); (c) @ = 20,340 mGal,theelevationsaretakenrelativetoaspherewithpi5) = 4909.7 km (isolinestakenatloo m intervals).

labelled gh'), that is gh') = 81,359 mGal, will take. The elevation isolines of this surface with respect to a sphere with radius p f ) = 2454.82 km are shown in Fig. 2.3b. Another set of isolines was constructed in order to represent a surface with $2) = i$$ = 20,340 mGal with respect to a sphere having radius po = 4909.7 km (Fig. 2.3~).In a similar manner, surfaces of equal radial gravity gradient (d2g/dp2), were derived for the same distances from the Moon's centre of mass: ph2) = 2454.82 km and pb3) = 4909.7 km. The elevations of the equal gradient surface ( m / d p z = (ag/dp)!,l) = 662.8 x s-l) vary within 3000 m with respect to a sphere with pb2) (Fig. 2.4b),this variation being already within 300 m with respect to a sphere with pb3)(d2W/47'= 82.8 x lo-' s - ~ )(Fig. 2.4~).If the range of the elevations of the equal radial gradient surface was greater by one order of magnitude than that of the elevations of the equal gravity surface near the lunar surface, at ph,) = 4909.7 km they would become the same. This is

Normal and Anomalous Gravitational Fields

125

further proof that the anomalies of the radial gravity gradient diminish rapidly as the distance from the Moon increases. 2.8 Anomalies of the Lunar Gravitational Field

The anomalous part of the gravitational field results from subtraction of the normal component from the observed gravitational field of the Moon. It is due to the uneven density inside the Moon and irregularities of its future. It is precisely the anomalous part that arouses the greatest interest at the current stage of gravitational studies of the Moon. The natural evolution of science implies establishment, at the initial stage, of the general pattern of a phenomenon. Advances in measurement techniques and instrumentation open up new possibilities for investigating the “fine” structure of an object or phenomenon. Anomalies of the lunar gravitational field give a new insight into the internal structure and figure of the Moon. Definition of the gravitational anomalies (gravity, gravity potential, gravity gradient, etc.) is a

126

Lunar Gravimetry

rather conditional operation in the sense that the magnitude of the anomalies depends on the manner in which the normal field has been defined. The normal part may include some or other harmonics of gravitational field expansion, or else the normal Moon is regarded as an ellipsoid of rotation, a triaxial ellipsoid, or a sphere. All these factors are taken into account in using and interpreting particular gravitational anomalies. We have already mentioned, in different contexts, the perturbing potential T(p,cp, A) which is defined as the difference between the real gravity potential (2.2.1) and the normal gravity potential (2.5.1): In the general form, all gravity anomalies can be written as: where y is normal gravity, g is the observed (measured) gravity, and 8, is a reduction correction whose structure determines the nature of the anomaly,

127

Normal and Anomalous Gravitational Fields

90.1 -180"

I

-120"

-60"

I

0"

60"

I

120"

I

180"

Fig. 2.4. lsolinesofequal radial gravitygradientG/dp2surfaceelevations (a) m / d p z = 1874.8 E; the elevations are taken relative to a sphere with pi') = 1738 km (isolines taken at 5000 m intervals). (b) m a p z = 662.8 E;theelevationsaretakenrelativetoaspherewith p,") = 2458 km (isolinestakenat1 00 in intervals). (c) d2W/dpl = 82.8 E;theelevations?wtakenrelativetoaspherewithpp' = 4909.7 km (isolines taken at 100 m).

Ag, of interest. In some cases, 6g takes care of the effects of the relief, while in others it enables reduction of the measured gravity to another surface. The normal gravity y(p, cp, A) at random points in space can be calculated using formula (2.6.1). But if the normal gravity y(cp, A) (2.6.3) corresponding to the normal Moon's surface is involved, some additional reductions are required. Let us represent the quantity y ( p , rp, A) as a power series expansion in terms of elevations H over the normal Moon's surface and restrict ourselves to the first three terms of the expansion

where -y(A, cp) is the normal gravity on the normal Moon's surface, (ay/i3p)o = a* W / a p 2 and ( f i / a p 2 ) o = m / a p 3 are the first and second radial

128

Lunar Gravirnetry

gradients of normal gravity (or first and second normal radial gradients of gravity). Their values have already been given in Table 2.1. In the case of a sphere whose radius is Ro = 1738 km, ($)o

=

-0.1874 mGal m-l

(3) 8p2 = 324 x

(2.8.3) mGal m-2

Remember, for comparison, that the mean vertical gradient of terrestrial gravity is -0.3086 mGal m- ’. In spite of the Moon’s mass being 81.3 times smaller than that of the Earth, the vertical gradient of lunar gravity is only one and a half times smaller than that of terrestrial gravity because of the Moon’s shorter radius. The so-called free-air gravity anomalies can find the widest aplications. They are derived as the difference between the measured and normal gravities

129

Normal and Anomalous Gravitational Fields

60". 200

90" -180"

---

-120"

(C)

at the same point M P , CPY

4= g(P9 CP, 4 - Y(P, CP, 4

= g(P, CP,1) - CY(CP, A) - 0.18748 162 x 1 0 - 9 ~ 2 1

+

where H is given in metres, while Ag, g and y are given in milligals. In studying the distribution of anomalous masses inside the Moon, one must exclude the gravitational effect of all masses except for the anomalous masses of interest. First of all, it is necessary to take into account the effect of the irregularities of the Moon's physical surface. Toward this end, an imaginary horizontal plane is drawn through the observation point. Since the most tangible effect is produced by the relief in the immediate proximity to the observation point, the gravitational effect due to the excess of masses above and their shortage below this plane is calculated. In either case, the masses tend to lower the observed gravity and, therefore, the correction for the gravitational effect of the relief is always positive. Thus, introduction of the correction for relief apparently eliminates the shortage of masses between

130

Lunar Gravimetry

the physical surface and the plane and “neutralizes” their excess above the plane. Now, at each observation point, the anomaly appears to be created by an infinite plane layer of masses, whose thickness depends on the elevation of the relief above normal Moon’s surface. The attraction of such a layer having thickness H and density a at a point whose elevation above the upper surface of the plane layer is h can be derived through evaluation of the following integral: (H [r2

6gn(0,0, h) = G o 0

0

0

+ h - z)r d+ dr dz + (H + h - z)2]3’2

= 2nGaH

(2.8.4)

Here, a cylindrical system of coordinates r, t,b and z is used, with its origin on the lower surface of the plane layer and the z axis passing through the observation point. The lower surface of the layer is taken at a level coinciding with the normal Moon’s surface. Expression (2.8.4) reveals an interesting property of the plane layer’s attraction. This attraction turns out to be independent of the distance h between the observation point and the layer, but depends only on the layer thickness H and density a. Substitution of the numerical values of n and G into the right-hand side of (2.8.4) gives 6g,

= 0.0418aH

(2.8.5)

where a is expressed in g ~ m - H ~ is, in m and 6g is in mGal. The gravity anomaly which includes, apart from the free-air correction, a correction factor excluding the attraction of the plane layer of masses between the physical and normal surfaces of the Moon, is known as Bouguer’s anomaly: AgB(p, ~ p ,A) = g(p, ~ p ,A) - y(v, A) - 0.1871H

+ 0.0418aH

(2.8.6)

It is named after Pierre Bouguer, the French scientist who used this anomaly in gravimetric studies during the famous expedition to South America in the middle of the 18th century. In principle, Bouguer’s anomalies must represent the distribution of only anomalous masses inside the Moon, because they take into account the elevations of the observation points and the effects of the masses, lying intermediate between the observation points and the normal Moon’s surface. Widely used in gravimetric studies of the Earth‘s figure are so-called mixed gravity anomalies. Although they are not so important for the Moon, mentioning them would be appropriate here. They are different from free-air anomalies in that the measured and normal gravities are associated with different surfaces, the measured gravity being related to the selenoid surface and the normal gravity to the normal Moon’s surface, as

131

Normal and Anomalous Gravitational Fields

follows: A.Sin(P, cp,

4= S(P, cp, 4- Y(V, 4

(2.8.7)

As regards the anomalies of the second gravity potential derivatives, the most interesting ones are those of the radial gravity gradient, which will be defined as the difference between the observed and normal radial gravity gradients: ’

a9 T -- (1874 - 0.0022H) x lo-’ s - ~ pp

- ap

(2.8.8)

(The second term in brackets takes into account variations in the normal gradient with height H reckoned from a sphere with R = 1738 km and given in kilometres.) The anomalies of the second radial gravity gradient may be determined from the formula TPPP

=

aFg apz - ($)o

(2.8.9)

In view of the smallness of (a2y/ap2)o(2.8.3), the anomalies Tpppare virtually equal in value to a2g/ap2. When the second and third potential derivatives are involved, the effect of the relief becomes substantially more important. The plane-parallel uniform layer exerts no influence on them. On the Moon, as opposed to the Earth, the anomalies of the second gravity potential derivatives are quite pronounced. Judging from the detailed measurements of line-of-sight accelerations from Apollo and gravity profile measurements from the lunar roving vehicle of Apollo 17, the anomalies of the second derivatives are as high as hundreds of eotvos (lo-’ s-~).

2.9 Relation between the Coefficients of Expansion of Different Parameters of the Moon’s Gravitational Field and Figure

Let us now see how the generalized spherical functions of the same degree for various parameters of the Moon’s figure and gravitational field are interrelated. First, we shall determine the amount 6 of difference between the radial gravity potential derivative a W p p and gravity g (2.3.2) at the same point:

6 = aW(P, cp, aP

4 - 9(P, c p , 4

=

-

i/p2(a w/acp)Z

+ i/p2 C O S ~cp(a w/an)2 WPP)

The numerator of the right-hand side includes squares of the horizontal attraction components tangential to meridians and parallels. Assuming, for

132

Lunar Gravimetry

example, that each of them has the highest possible value of 500 mGal, it can be easily calculated that 6 = 0.25 mGal. Thus, virtually every conclusion drawn as regards the value of a W/ap fully applies to gravity g. Using the expansions of the lunar gravity potential (2.2.1) and radial gravity potential derivatives (2.2.2), (2.2.3) and (2.2.Q and ignoring the nonlinear and centrifugal-tidal terms in the latter, we shall compose a table of factors (Table 2.5). Given the coefficients of expansion of a radial derivative of the lunar attraction potential in this table, one can easily determine those of another derivative. To do that suffices to multiply the harmonic coefficients of the first expansion by the respective tabulated factors. We shall henceforth refer to these as transformation factors. For instance, the harmonic coefficients g,, and h,, of gravity expansion are given. In order to determine the harmonic coefficients K,, and M,, of the expansion of the radial gravity gradient d2 W/ap2,it is necessary to multiply, respectively, g,, and h,, by the transformation factor - (n + 2)/p according to the table. Conversely, given the values of K,, and M,,, to determine g,, and h,, the former must be multiplied by transformation factors - p/(n + 2). But since the transformation factor for any radial potential derivative is independent of m, virtually the entire sum over m is multiplied by the transformation factor-that is, a generalized spherical function representing the derivative under consideration. Consider now the relations between some characteristics of the Moon’s figure and gravitational field anomalies. The perturbing potential T is related TABLE 2.5 Transformation factors for the spherical harmonics of different radial attraction potential derivatives

U

V

V

I

av -

n+l --

aP

a2v -

aP3

aP

I

P

(n

aP2 a2 v -

av -

+ l)(n + 2)

n+2 --

P2

- (n

+ l)(n + 2)(n + 3)

P

(n

P3

+ 2)(n + 3) PZ

0,

1

n+3

--

P

I

= Ku,

where u is the initial field characteristic, 0 is the transformationresult and K is the transformation factor.

133

Normal and Anomalous Gravitational Fields

to the elevations [ of the selenoid above the normal Moon’s surface in a rather simple fashion. This relation stems from the property of level surfaces, mentioned in 51.3. In gravimetry, the term “Bruns formula” is applied to the relation T

[=-

(2.9.1)

Y

where, in the Moon’s case 7 is the mean normal gravity between the surfaces of the normal Moon and selenoid. If gravity anomalies Ag are determined from observations, rather than the perturbing potential T, equations through which these characteristics of the gravitational field are related are necessary. Let us write a formula expressing mixed gravity anomalies Ag,(p, cp, A) in terms of the perturbing potential T and its derivative aT/ap, both related to the normal Moon’s surface: (2.9.2) Now we can define the transformation factors between the principal spherical functions representing the figure and gravitational field of the Moon, namely, the elevation (2.9.1) of the selenoid above the normal Moon’s surface, mixed gravity anomalies Ag, (2.9.2),the perturbing potential T (2.8. l), and the radial derivative aT/ap (2.8.4) of the perturbing potential. To this end, the characteristic should be expressed as expansion in generalized spherical functions m

T=

OD

n=2

(2.9.3) (2.9.4)

Tn

(2.9.5)

C=

m

1

n=2

(2.9.6)

Cn

Introducing expressions (2.9.3)-(2.9.6) into equation (2.9.2), we find that n = 2 , 3 , ...

(2.9.7)

It is easier to determine the transformation factors between other characteristics, which can be done using expressions (2.8.9) and (2.9.3)-(2.9.6). Table

134

Lunar Gravimetry

TABLE 2.6 Transformation factors for the spherical harmonics of some anomalies of the Moon's gravitational field and figure U

($)"

n+l -

I

P

r

-1

P

I

(n + 1)Y

Y

,

u,,, = Ku,,,,,

where u is the initial field characteristic, 0 is the transformation result and K is the transformation factor.

2.6 summarizes the transformation factors for the generalized spherical functions of the anomalies of the Moon's gravitational field and figure. 2.10 Moon's Relief and Gravitational Field

The Moon's relief has a complex structure. It features vast highlands and plains. It is standard practice to represent the relief in the form of isolines of equal elevation of a smoothed physical surface of the Moon above a reference surface. The easiest way is to take a sphere as the reference surface, whose radius is assumed to be equal to the mean radius of the Moon. For numerical analysis, the relief can most conveniently be represented as an expansion in spherical or sample functions. The radius vector p(cp,A) of the smoothed physical surface of the Moon, written as an expansion in spherical functions, is 1

N

+n=l

n m=O

(a,,,,, cos mA + 6,,,sin mA)P,,,,,(sin cp)

1

(2.10.1)

where N is the order of the expansion, dependent on the validity and number of the initial absolute elevations of the lunar surface, and Ro is the radius of the reference sphere. The coefficients a,,,,, and b,,,,, are determined from the available coordinates of points on the Moon's physical surface. If we go to

135

Normal and Anomalous Gravitational Fields

elevations h of the relief above the reference sphere, we have N

n

C n= 1 m=O

RO 1

(ii,m

cos m l

+ 6,, sin ml)Pnm(sincp)

(2.10.2)

where in, and Fnmare normalized coefficients, and Pnm(sincp) stands for normalized Legendre functions. Approximation of the relief by an expansion in spherical functions of the type shown in (2.10.2) inevitably involves two kinds of errors. One of these is due to the finiteness of the series, while the other stems from the expansion coefficients. Let N be the highest order of the retained terms in the expansion. Then, the first of the two errors is m

el = Ro

n

C C n=N+1

m=O

(anm cos m l

+ b,,

sin ml)P,,(sin cp)

If the errors in the expansion coefficients are labelled Aanmand Ab,,, the second error is e2

= RO

N

n

1 1( fAanmcos mA -L Ab,,

The overall’varianceof

8’ m

E’

sin ml)P,,(sin cp)

n=O m=O

=

1

n=N+l

equals the sum of the variances of Dn

E:

and

8;:

+ nC= O d n = D - C (Dn-dn) N

N

n=O

where d, stands for the degree variances of the errors involved in the determination of the coefficients an, and b,,, D , stands for the degree variances of the coefficients anmand b,, themselves, and D is the variance of the Moon’s relief elevations. The latter equation shows that it makes sense to increase the order N of the expansion by one each time Dn+ - d, + > 0;that is, if Dn+ > d,+ 1. Thus, the optimal order of the expansion depends on the ratio between the degree variances of the errors involved in the determination of the coefficients and those of the coefficients themselves. The first expansion of the Moon’s relief in spherical functions up to the eighth order was carried out by Goudas (1968) who used absolute elevations of the near-side relief. The total lack of data on the Moon’s far side has given rise to certain assumptions. Goudas proceeded from symmetry of the relief on the near and far sides. This is why his expansion could adequately describe the near-side relief and have nothing to do with the far-side relief. Since the first expansion of the Moon’s relief, more data on the latter have become available. The early selenodetic catalogues of absolute elevations on the near side were revised and compiled catalogues have been prepared (Gavrilov et al., 1977; Lipsky et al., 1973; and others). The elevations were

136

Lunar Gravimetry

reduced to the Moon’s centre of mass. The basic methods for compiling unified catalogues have been outlined (Gavrilov, 1969; Lipsky et al., 1973). The principles of establishing base networks and their relative deformations, techniques of lunar polygonometry and triangulation have also been described (Gavrilov, 1969; Gurshtein and Slovokhotova, 1971; Habibullin et al., 1972; Light, 1972; Helmering, 1973). The most serious drawback of the available elevation data is the limited knowledge about absolute elevations on the far side. This drawback has begun to be remedied by satellite observations, which have been instrumental in the determination of the selenodetic longitudes, latitudes, and absolute elevations of several fundamental base points. In particular, Wollenhaupt et al. (1972) used the results of numerous optical measurements from Apollos 8, 9, 10, 11, 12, 14 and 15 to determine 31 base points along the equatorial belt of the Moon. Their latitudes range from - 1 1 ” to + 26”;21 out of these points are on the near side and 10 on the far side. The coordinate determination errors are 0.7 ( q ) and 0.6 (A) km, while those of absolute altitude determination do not exceed 0.4 km. The coordinate errors were determined by the accuracy of location of the Apollo spacecraft. Extremely valuable data on absolute elevations were obtained by laser altimetry from Apollos 15, 16 and 17 (Wollenhaupt and Sjogen, 1972; Kaula et al., 1973, 1974; Sjogren, 1977). Elevations of the physical surface were measured along profiles (Fig. 2.5) extending across the entire Moon in its equatorial zone. They are given on the profiles relative to a spherical Moon with a radius of 1738.0 km. The centre of this sphere coincides with the Moon’s centre of mass. Laser measurements were taken from the orbiting Apollo spacecraft at 20 s intervals, which corresponds to 30 km of covered distance. Although the altimeter was sensitive to within about 2 m and the spot of the laser beams on the lunar surface was only about 30 m in diameter, there was no need to probe at shorter intervals because the uncertainty in the Apollo’s coordinates remained considerable. The position of the spacecraft was determined from the Earth by Doppler tracking at 10 s intervals. For intermediate points in time, the position of the spacecraft on its orbit was computed using a model of the lunar gravitational field. The uncertainty in the Apollo’s position in the direction of the laser beams did not exceed an absolute value of 0.4 km, while the error in the relative position between measurements was 0.1 km. Laser altimetry has made it possible, in addition to studying the Moon’s relief, to solve a number of other problems. In particular, the position of the centre of the Moon’s figure with respect to its centre of mass was determined more accurately and the geometrical figure of the Moon was defined more exactly. Some common features of the Moon’s relief were established. Table 2.7 lists the mean elevations of some features on the Moon’s near and far sides

-E *

1

re

4

o

U

-4 -81

I

90"

I

I

120"

I

I

150"

I

I

180"

I

210"

I

1

240"

I

I

270"

I

1

300"

1

I

330"

I

I

360"

I

I

30"

I

I

60"

I

I

90"

I

8-

-

-E @ Y U

1

:i-flkg+q% --4

-

-

-----

------

#

------

Fig. 2.5. Intervals in elevations of the physical surface on the near and far sides of the Moon, relative to a spherical Moon with radius R = 1738 km, measured by laser altimetry from Apollo 15 (top) and Apollo 16 (bottom) (Kaula eta/., 1973, 1974; Siogren and Wollenhaupt. 1972, 1973).

138

Lunar Gravimetry

TABLE 2.7 Mean elevations of some formations on the Moon, relative to a sphere with radius R = 1738 km (Kaula ef a/., 1973)

Mean elevation according to (km) Formation type Far-side highlands Near-side highlands Circular maria Other maria

% of

Apollo

surface

15

57 23 6 14

+ 1.9 - 1.7

-4.1 -2.0

Apollo 16

Apollo

f2.1

+0.9 - 1.3 -3.7

- 1.2

-4.1 -2.5

17

-2.1

Weighted mean elevations (km)

+ 1.8

- 1.4 -4.0 -2.3

above a sphere with R = 1738 km, determined from these measurements (Kaula et al., 1973). The tabulated elevations belong to the equatorial zone and are not adequately representative of the Moon as a whole. The elevation profiles constructed from laser altimetry data (see Fig. 2.5) attest to the qualitative difference in relief between the near and far sides of the Moon. The maria have smooth floors inclined from west to east at about 1 : 500 to 1 :2000 (Kaula et al., 1973). An inverse correlation is observed between the depths of circular maria (Mare Serenitatis, Mare Crisium, Mare Smythii) and their ’diameters. The interferometric measurements with the aid of Earth-based radars, which preceded laser altimetry, allowed determining the elevations of only large features of the lunar surface, their accuracy being in the neighbourhood of 200 m (Zisk, 1972). The first data on the far-side relief elevations in the western hemisphere were provided by the photographs taken from the automatic probe Zond 6. They revealed a wide depression on the southern far side (Rodionov et al., 1971, 1976; Ziman et al., 1975). The level of the physical surface over a sizeable area (Fig. 2.6) was about 4.7 km below the mean level of the Moon’s surface. The upland region in the south, having a relative elevation of about 2.6 km, extends into Mare Australe on the near side. Satellite data on the geometrical figure of the Moon cannot substitute for the knowledge accumulated over more than fifty years of astronomical investigation of lunar surface elevations. Yet they provide the only insight into the Moon’s far side. It was extremely important to obtain information on the overall geometrical figure of the Moon with due account for both the near and far sides as well as its relationship with the centre of mass, which has been partially accomplished by way of laser altimetry. The results of the photogrammetric processing of the pictures taken from Apollos 15,16 and 17 are going to be used in establishing a highly accurate base network covering

139

Normal and Anomalous Gravitational Fields

0"

r

+

Fig. 2.6. The Moon's physical surface elevations, in a section from a plane near to 2. = 180", from the photographs taken from Zond 6 with a 20-fold magnification (Rodionov et al., 1971).

20% of the lunar surface (including its far side) and having a density of one base point per 900 km2 (Light, 1972; Helmering, 1973). The accumulated elevation data have made it possible to achieve more or less reliable expansions of the lunar relief in spherical functions. Bills and Ferrari (1977) handled in their work the results of 5800 laser altimetric measurements, 1400 photographs taken from Apollo spacecraft with an accuracy of f0.3 km, and 3300 elevation measurements based on photographs taken from the Earth. The latter were said to be accurate to within f 1.0 km. The terrestrial measurements were corrected for displacement of the Moon's centre of mass by 1.77(f0.16) km toward a point with coordinates 25"s and 191"E. The elevations were determined with respect to a sphere with R = 1737.46 km. The map of elevations of the smoothed Moon's relief, based on the work by Bills and Ferrari (1977), covers the zone confined within zk 45" latitude. The elevations vary from +5.5 to -2.5 km; that is, the total variation range is 8 km. Proceeding from the up-to-date lunar relief elevation data, Chuikova (1975a, 1975b, 1978) expanded the relief. She used a hyposometric map of the

140

Lunar Gravimetry

Moon’s near side confined within f70” latitudinally and logitudinally, based on Mills’ catalogue, the coordinates of the above-mentioned 3 1 base points (Wollenhaupt et al., 1972), the results of laser altimetry from Apollos 15 and 16 (Sjogren and Wollenhaupt, 1973), the absolute elevations of 68 points of the liberation zone on the far side in the Moon’s western hemisphere, determined from space probes Zond 6 and Zond 8 (Ziman et al., 1973)’ the elevations determined with the aid of Zond 6 along the far-side meridional profile (Rodionov et al., 1976), as well as the catalogue of elevations in the peripheral region, compiled at the Main Astronomical Observatory of the Ukrainian Academy of Sciences of particular interest were the data provided by Zond 6 and Zond 8; they have not been used in other works. This is precisely why on the averaged relief map drawn in this work (Fig. 2.7) the regions in the southern part of the western hemisphere, polar regions, and the peripheral region better represent the real Moon. Consider now the degree variances, not of the relief elevations themselves, but of their horizontal gradients along tangents to meridians and parallels. The expressions for these degree variances as applied to the relief will take the forms n (Dh,)n =

+ Br%)

m=O

(2.10.3) The coefficients A,,,, Brim, R,, and fin,,,can be computed according to the formulae (3.3.3) where the coefficients a,,, and 6,,,of expansion of the relief Figure 2.8 presents graphs elevations must be substituted for Cnmand Snm. showing the degree variations in r.m.s. values ahq = [(Dhq)n/(2n and a h l = [(Dh,),J(2n 1)]’/2 calculated using the relief expansion coefficients (Chuikova, 1978). Autocorrelation functions of the Moon’s and the Earth‘s reliefs are shown in Fig. 2.9. Let us examine the anomalous gravitational field due to the Moon’s relief. The masses constituting the visible relief will be represented condensed on a sphere’s surface as a simple layer with surface density

+

+

On(%

4= aowcp, 4= aoR

N

n

1C n=l

m=O

(anm

cos mA

+ bn, sin ml)P,,(sin

cp)

(2.10.4)

where a. is the mean density of the relief-forming (2.10.4) rock. The gravity potential of the masses in the simple layer is V(P, cp,

4=G

jj S

CP,1)dS

r

Fig. 2.7. Elevations of the lunar relief relative to a sphere with radius R = 1738 km (isolines taken at 0.5 km intervals)

r;

40'

c

I I

'E Y

E 30 E

I

I

IIII

20

n Fig. 2.8. R.m.s. values of the spherical harmonics of the horizontal relief elevation gradients along meridians and parallels plotted against position in degrees. (1 ) (uhJn along meridians; (2) ( u , , ~along )~ parallels.

143

Normal and Anomalous Gravitational Fields

where I is the distance between a current point on the surface and the point (p, cp, A) at which the potential is considered. Substitution of the expansion for l/r (1.3.2) and that for 6, (2.10.4) into this equation gives V(p, cp,

A) = GaoR

ss

c N

n

1(anrncos rnl + 6,, sin rnA)Pnm(sincp) n = 1 m=O

S

In expansion (1.3.2),it was assumed that p1 = R. By resorting to the theorem of restoration of spherical functions (Idelson, 1936), we can simplify the righthand side of the latter equation:

x

(anrncos rnl + li,

sin rnA)Pnm(sincp)

(2.10.5)

Comparison of the coefficients of the resulting expansion with the corresponding ones of expansion (2.3.4)gives (2.10.6) Introducing the Moon’s mean density au instead of its mass Mu, we can of the gravitational potential expansion to relate the coefficients Cnmand Snm

Fig. 2.9. Normalized autocorrelation functions KOn($) of the (1) Moon’s and (2) Earths reliefs.

144

Lunar Gravimetry

the corresponding coefficients an, and 6,, of the relief expansion in the following manner: (2.10.7) Assuming that the Moon’s density varies with depth as a(p) = 0 0

+ upp

where u and p are constants governing the density variation pattern, instead of formula (2.10.7), we have (Goudas, 1968, 1973): (2.10.8) Assuming that u = 0, we have (2.10.6) instead of (2.10.8), which corresponds to the relief density being constant and equal to the mean density of the Moon. Given in Table 2.8 are the values of normalized harmonic coefficients C,,, and S,,, of the gravitational potential, calculated using formula (2.10.6) based on the coefficients of the Moon’s relief expansion executed by Chuikova (1975a,b) as well as by Bills and Ferrari (1977). This table also lists for comparison the harmonic coefficients derived by Ferrari (1 977) and Akim and Vlasova (1977) from the perturbed motion of the ALS. The values of en, and S,,,, corresponding to the relief, are much higher. Also compare the degree variances of the gravity potential calculated with reference to the Moon’s relief and determined from tracking of the perturbed motion of the ALS (Fig. 2.10). The first degree variances are much greater than the corresponding variances determined from the perturbed ALS motion. The TABLE 2.8 Comparison of the normalizedharmonic coefficientsof the gravitationalpotential, calculatedfrom the observed Moon’s relief and determined from ALS tracking data ( x

Go

Gl

From relief (Chuikova, 1975)

-607.6

-603.8

From relief (Bills and Ferrari, 1977)

-212.3 f25.8

-605.8 f17.5

Determination

Go

Ctl

G 2

-21.9

-119.3

-78.98

87.23

-147.5 f13.6

-135.9 f22.1

-149.8 f26.5

11.5 f5.7

-91.52 51.7

2.04 f1.8

f 1.8

-90.03 f 1.0

0.7 f1.4

35.47 f0.5

&I

From ALS tracking data (Ferrari, 1977) From ALS tracking data (generalized model)

0.58

f 1.7

1.97 f9

1.3 k1.4

33.66

145

Normal and Anomalous Gravitational Fields

n

Fig. 2.10. R.m.s. values of the spherical harmonics ( u T ) , for the lunar gravitational potential plotted against degree, n. (1 ) Calculated from the Moon's relief; (2) determined from ALS tracking data.

difference diminishes as the order n of the harmonics increases. This pattern can be explained by the different degree of isostatic compensation of masses differing in extent. The relief masses over small areas cannot overcome the strength of the lunar crust and remain isostatically uncompensated, which is to say that they are responsible for a stronger gravitational field. As regards regions characterized by low harmonics, the isotatic compensation is virtually complete; that is, corresponding to excess of relief masses is a

c30

€31

c32

€33

$3 1

67.6

-9.i~

-16.5

63.6

102.4

47.1

23.5 k19.3

167.0 k14.6

62.2 k20.5

102.1 k28.0

47.2 k18.1

250.1 k11.1

53.1 - 18.2

+

51.9 k22.7,

28.9 k11.4

-0.23 k2.3

0.10 k2.2

-4.90 k4.5

27.04 k1.5

11.52 k3.8

24.98 k7.1

6.00 f1.7

1.18 k4.7

-9.02 & 6.8

1.26 k1.7

-1.97 k0.7

3.5 k1.6

23.29 k1.2

12.67 k0.8

k0.5

15.82

* 6.16 1.1

6.53 f0.9

-8.44 k0.4

$21

$22

54.0

1.73

-7.35

146

Lunar Gravirnetry

deficiency of mass below and vice versa. As a result, the anomalous gravitational field approaches zero, as can be seen on the grounds of Fig. 2.10. It can be inferred from Table 2.8 that the geometrical figure of the Moon cannot be derived from the coefficients of expansion of the lunar gravitational field determined from perturbations in the ALS motion. This is precisely what Volkov and Shober (1969) did following Goudas (1968, 1973). Disregarding the internal mass distribution will not produce anything close to the actual relief of the Moon.