Geophys. J. R . astr. SOC. (1979) 57,639-648

Hydrostatic figure and related properties of the Earth

S .M. Nakiboglu

Surveying Department, University of Queensland, St Lucia, Queensland 406 7, Australia

Received 1978 October 31; in original form 1978 September 11

Summary. The second-order internal theory of hydrostatic equilibrium initially developed by Kopal is extended to form equations equivalent to those of de Sitter. This new development is used together with the density distribution of Dziewonski, Hales & Lapwood to compute flattening, precessional constant and moments of inertia of the hydrostatic earth. These values are compared with those of the real Earth. Introduction The internal equipotential surfaces of a self-gravitating, rotating mass in hydrostatic equilibrium can be determined by observing that the equal-density and equipotential surfaces coincide within the body. The spherical harmonic coefficients of the series expansion of the equation of these surfaces, as well as the density, are functions of one independent variable only, namely the mean distance from the centre of mass of the body. Then, the volume integrals giving the spherical harmonic coefficients of the internal gravity potential can be transformed into a boundary-value problem composed of a system of second-order ordinary differential equations, satisfied by the harmonic coefficients of the equipotential surfaces, and their associated boundary conditions given at the centre and the surface of the body. The first-order theory yielding the second degree zonal harmonic coefficient was established by Clairaut in 1743. Darwin in 1900 and de Sitter in 1924 developed the secondorder theory which includes the fourth-degree harmonic (Jeffreys 1970, pp. 183-192). Kopal (1960) introduced a systematic treatment of the second-order theory which was later extended into the third-order one (Kopal & Lanzano 1974; Lanzano 1974). Kopal’s theory has definite advantages over the previous ones due to its systematic and explicit development. Boundary-value problem for the interior field Let the equation of an internal equipotential surface be ca

n=O

S. M . Nakiboglu

640

where is the mean radius of an internal equipotential surface expressed in terms of the mean radius of the outermost surface R as a unit, 8 is the complement of the geocentric latitude, P, is the nth degree Legendre polynomial and f,(@ is the unknown zonal-harmonic coefficient. We need t o consider the terms for n G 4 in the second-order theory. f2 is of the order of flattening and the remaining coefficients are of O(f:). The coefficient fo is related t o fi as fo= -f:/5 thus leaving fi and k t o be determined. Kopal (1 960, p. 70) and James & Kopal (1963) give the boundary-value problem for fi and k.The governing differential equations and their associated end conditions are

P’f;’

6

t 6 - (of;

D

f2)

- 6fz =

8f; (of;

6 9 f i ) - 9 - Pf;

D

(of;

2h)]

where

p(o)

is the density,

$0)

=

23

p = p” j P

=

I0 0

pp’ do is the

mean density within surface

0,

is the mean density of the Earth,

m = u2R3/GMis a dimensionless quantity of O ( f i ) , w , M a n d G being the rotation rate and mass of the Earth and the gravitational constant respectively. The equations (2-6) are equivalent to those derived by de Sitter and Darwin for ellipticity E and deviation, K , of equilibrium figure from an ellipsoid of revolution (James & Kopal 1963). The equation of an ellipsoid of revolution with semi-major axis a and flattening E is r = a( 1 - E cos’ 8 - 3/8

E’

sin’ 28).

Then the equation of equilibrium figure which deviates from the ellipsoid in second-order terms is written as

r = a [ I - E COS*O- (3/s E’

tK)

sin’ 281.

Hydrostatic jigure of the Earth

64 1

The comparison of this with equation (1) shows that

35

K

=-

27

f4

--

32

f;.

32

Bullard (1946) solved the boundary value problem for E and K for the Earth initially developed by Darwin and de Sitter (1924) (see Jeffreys (1953) for corrections to de Sitter’s and Bullard’s formulae). James & Kopal (1963) solved the problem as given in equations (3) t o (6). Both investigations used Bullen’s density model. Jeffreys (1963) obtained the solution by making corrections t o the solution of the first-order Radau’s equation using a simplified density model. Nakiboglu (1976) solved the problem using a recent density distribution of Dziewonski, Hales & Lapwood (1975) (Fig. 1). Table 1 gives the results of these computations. The comparison of the results shows reasonable agreement for E in spite of the large differences in density models especially around the centre of the Earth.

6/0

0.0

0.2

0.2

0.4

1.0

0.8

0.6

-

P 0.6-

I

0.0

I

0.5

I

1

I

.o

I

I

I

I

2.0

1.5

I

I

2.5

Figure 1. Variation of 6/Dand D within the Earth.*

Table 1. Source

€-I

K

lo*

E

and K derived from internal theory. Bullard (1946)

Jeffreys (1963)

James & Kopal (1963)

Nakiboglu (1976)

297.338 68

296.75 64

296.8 65

296.961 46.4

The internal theory, together with the knowledge of p , w and R , is sufficient to determine the moments of inertia, precessional constant and the exterior gravity field as well as the surface of the equilibrium figure. These quantities are computed and compared with those of the real Earth in a previous work (Nakiboglu 1976). It is sufficient to note that there are significant discrepancies between the observed and equilibrium values of the precessional constant H , the second-degree zonal harmonic coefficient of gravitational potential J2 and moments of inertia. These discrepancies are the measures of the deviation of the Earth from hydrostatic equilibrium. However, parts of the discrepancies are obviously ‘Obtained from the density distribution of Dziewonski e l al. (1975).

S. M. Nakiboglu

642

due to the observational errors in p , J2 and H. Probably J2 and H are the most accurately known quantities. Therefore, using the accurately known J2/H as the hydrostatic value and then computing J2 and H from the hydrostatic theory will yield the most accurate measure of the deviation of the Earth from hydrostatic equilibrium. The comparisons of the hydrostatic values of J2, H and ellipticity E with accurate observations should yield some information on the minimum strength of the Earth. The internal theory of Kopal (1960) summarized above is extended in the following sections t o develop the hydrostatic theory incorporating the observed J2/H. Moments of inertia and related quantities Moments of inertia of the equilibrium figure about the polar and equatorial axes are derived by Kopal(l960, p. 85):

We obtain the second-degree zonal coefficient of the gravitational potential, J 2 = (C -A)/Ma*, by combining the preceding equations with 1

3

1

23

Hence,

q = C/Ma2can be found similarly as

The equations (8) and (9) can be combined to yield

It should be noted that the similar expression of de Sitter (1924, equation 19) omits the terms of o(E'). The integral in equation (10) can be put into a more convenient form, independent of fi. The first term, after integration by parts, becomes

Defining Radau's parameter as

v'=-Of; f2

1 +2f2

1 +2f2 =7?2

~

1+f2

- 7

1+f2

Hydrostatic figure of the Earth

643

equation (1 1) is transformed into (1 3)

where

is a small quantity of the order of

which is the deviation of the mean value of

over 0 g fl G 1 (Bullard 1946). The second term within the integral in equation (10) can be evaluated using equations (3.39) and (5.10) of Kopal(l960, pp. 59-68) yielding

where

(-

1 1 1 5 1 a = - - - in D p .). 9 5 3 Finally, equation (10) is combined with equations (13) and (I 5) to obtain

5

l t h

All values in equation (16) refer to fl = 1.

Exterior gravity field Since the surface of the equilibrium figure is equipotential the exterior gravitational potential V is the solution of the boundary-value problem stated as

V2 V = 0 outside the body V = UO- 34 u 2 r 2 sin20 on the boundary surface, where Uois the geopotential number on the surface. The solution is obtained by observing symmetry properties and using equation (1).

S. M. Nakiboglu

644 where

m

f2---

J2=-

3

11

-f

Z2

1

- - mf2,

7

7

36 6 J4=- f 4 - -f i - - m f z , 35 7

u,=-

2

1 + - - - - + - m f 24) .

GM( R

15

The harmonic coefficients Jznare related to those of de Sitter by the relations J2=(2/3)J and J4=- (411 5 ) K .

Computations and results The basic equation is obtained by combining equations (12), (1 6) and (1 8),

Y l f i + Y 2 f 2 + 7 3 = 0, where y1 = -

y2=-

15

117

723 q2+ - 4 40 140

-

3 1--q 2

2

1

-

-

7 '

4 +m('--q), 2s (1+h)2 21 7 1+2772

4

--

~

Equation (19) together with the boundary conditions on f2 and k (equations (5) and (6)) should yield the solution once the density distribution within the Earth is specified. The following values of the quantities related to density are obtained by numerical integration using the density model of Dziewonski et al. (1975).

/01Dp4dfl= 0.25184805, ff

= 0.094977

,

The value (f4'/f4)lp = = 0.63720565 is taken from Nakiboglu (1976). The quantities q = 0.33071474 and m = 0.00344980146 are obtained using Jz= 0.001082641, w = 0.72921 151467 10-4rad s-', G M = 3.986003 10i4m3s-', R= 637 1009.2 rn (Nakiboglu 1978) and H = 0.00327364 (Khan 1969). Initially equations ( 5 ) , (6) and (20) are solved simultaneously, taking X = 0 which gives

645

Hydrostatic figure of the Earth Table 2. Hydrostatic values as obtained by various authors. Source

€-I

Henriksen (1960) Jeffreys (1 96 3) Khan (1967) Present values

300.0 299.67 299.15 299.829

K

lo6

A 10,

q’

0.5587

-

0.64 1.409

-

0.5882100

-

1.3 3.98

Table 3. Comparison of real and hydrostatic values.

Real value Hydrostatic value

7)’

€-I

J, lo6

-J, lo6

Alma2

H 103

298.2564 299.829

1082.641 1074.467

0.547 3.636

0.3296321 0.3296403

3.21364 3.24892

= 0.586948868. X is computed from equation (14) using the value of 77’ as h = 3.98 x

The procedure is repeated this time with the computed value of X. The results are: -2228.1064 x loF6, f4= 5.118 x qz=0.58952641, ~ ‘ = 0 . 5 8 8 2 0 9 9 5 , e-’= 299.8288 and K = 1.409 x 10 +. Table 2 shows a comparison of these values with the ones obtained by using de Sitter’s and Jeffreys’ approaches. The computed hydrostatic values of J,, J4, H and AIMR’ are given in Table 3 together with the corresponding values for the real Earth. The minimum strength of the Earth can be estimated from p = 4.214 AJ, 10”dyne cm-’ (Jeffreys 1963) with the assumption that the extra equatorial bulge is supported by the strength to a depth of 0.455R of the Earth. The result is p = 3.3 x 10’dyne cm-’. Stacey (1977, p. 247) gives the estimates of stresses associated with convection and viscosity in the mantle from the dissipation of mechanical energy in the motions of tectonic plates. The values are 0.8 x 10’dyne cm-’ and 3.2 x 10’dyne cm-’ respectively for an oceanic plate moving at 8 cm/yr and a continental plate moving at 2 cm/yr. The strength estimate in the present work confirms average plate motion of about 2 cm/yr. Another indication of the Earth’s strength is the comparative values of fluid and secular Love’s numbers. To a first-degree approximation we can write the fluid and secular Love’s numbers as

fi=

K, =

3H4 m(l -2/3e)

f

where eh denotes the hydrostatic flattening. Substituting relevant values here, we obtain K f= 0.934, k, = 0.944. The free-air cogeoids with reference to the hydrostatic figure and the Earth ellipsoid are shown in Fig. 2(a) and (b) respectively. The cogeoids are derived from the first 12 degree harmonic coefficients of GEM 8 Model (Wagner et al. 1977). The most striking difference between the two figures is that the hydrostatic cogeoid exhibits polar depressions and a distinct pear-shape. Since this cogeoid refers t o a figure of zero deviatoric stress, the undulations should correspond to the long-wavelength deviatoric stress distribution and lateral inhomogeneities within the Earth. The glaciation effects above 60” north and south latitudes are clearly visible.

646

S. M. Nakiboglu

Hydrostatic figure of the Earth

647

648

S. M.Nakiboglu

Comments The hydrostatic values derived from the internal theory (Table 1) and the modified theory of Kopal (Table 2) d o not agree. The simplest reason for the disparity could be the incompatibility of density distributions with the observed J2/H under hydrostatic assumption. The internal agreement of the values in Table 1 is to be expected because the surface value of E is almost independent of the interior structure over a wide range of possible density distributions (Jeffreys 1963). Acknowledgment The author is indebted t o F. D. Stacey for critically reading the manuscript. References Bullard, E. C., 1946. The figure of the Earth,Mon. Not. R . astr. SOC.Geophys. Suppl., 5 , 186-192. de Sitter, W., 1924. On the flattening and the constitution of the Earth, Bull. astr. Insts Neth., 2, 97108. Dziewonski, A. M., Hales, A. L. & Lapwood, E. R., 1975. Parametrically simple earth models, Phys. Earth planet. Znt., 10, 12-48. Henriksen, S. W., 1969. The hydrostatic flattening of the Earth, ICY Ann., 12, 197-198. James, R. & Kopal, Z., 1963. The equilibrium figures of the Earth and the major planets, Zcarus, 1, 442-454. Jeffreys, H., 1953. The figures of rotating planets,Mon. Not. R . astr. SOC.,113, 97-105. Jeffreys, H., 1963. On the hydrostatic theory of the figure of the Earth, Geophys. J. R . ash. SOC.,8, 196-202. Jeffreys, H., 1970. The Earth, 5th edn., p. 525, Cambridge University Press. Khan, M. A., 1969. General solution of the problem of hydrostatic equilibrium of the Earth, Geophys. J. R . astr. SOC.,18, 177-188. Kopd, Z., 1960. Figures of Equilibrium of Celestial Bodies, p. 135, The University of Wisconsin Press, Madison. Kopal, Z. & Lanzano, P., 1974. Third-order Clairaut equation for a rotating body of arbitrary density and its application to marine geodesy, NRL Rep. No. 7801, Naval Research Laboratory, Washington DC. Lanzano, P., 1974. The equilibrium of a rotating body of arbitrary density, Astrophys. Space Sci., 29, 161-178, Nakiboglu, S. M., 1976. Hydrostatic equilibrium figure of the Earth, Aust. J. geod. photogramm. Surv., 25, 1-16. Nakiboglu, S . M., 1978. Variational formulation of geodetic boundary value problem, Bull. Geod., 52, 9 3 -100. Stacey, F. D., 1977. Physics of the Earth, 2nd edn, p. 414, John Wiley & Sons, Inc., London. Wagner, C. A., Lerch, F. J., Brownd, J. E. & Richardson, J. A., 1977. Improvement in the geopotential derived from satellite and surface data (GEM 7 and 8), J. geophys. Res., 82, 901 -914.