A Marchenko - On the gravitational potential energy of the earth - страница 1

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Геодезія, картографія і аерофотознімання. Вип. 69. 2007

A. Marchenko

Lviv Polytechnic National University

ON THE GRAVITATIONAL POTENTIAL ENERGY

OF THE EARTH

© Марченко О.М., 2007

Для оцінювання гравітаційної потенціальної енергії Землі E використано ЗБ-розподіл густини еліпсоїдальної планети разом з його оцінкою точності. Саме використання останньої дало змогу виконати оцінювання E на основі лише радіального розподілу густини у вигляді її неперервних та кусково-неперервних моделей: Лежандра-Лапласа, Роша, Булларда і Гаусса. В результаті отримана нерівність для E з верхнею границею EH для однорідного розподілу і нижнею границею EGauss, яка відповідає розподілу Гаусса для густини Землі. Головні оцінки E дають гарне погодження з EGauss: як у випадку E, яке базується на моделі Роша з 6 головними стрибками густини, так і оцінки E, що відповідають 4 найпростішим моделям з одним стрибком густини на границі ядро-мантія.

Для оценки гравитационной потенциальной энергии Земли E использовано ЗБ-распределение плотности внутри эллипсоидальной планеты совместно с ее оценкой точностью. Именно применение последней дало возможность выполнить оценивание E только на базе радиального распределения плотности в виде ее непрерывных и кусочно-непрерывных моделей: Лежандра-Лапласа, Роша, Булларда и Гаусса. В результате получено неравенство для E с верхним пределом EH для однородной планеты и нижним пределом EGauss, соответствующим Гауссовому распределению. Главные оценки E дают прекрасное согласование с EGauss, включая значение E, основанное на модели Роша с 6 главными скачками плотности внутри Земли, и оценки E, отвечающие простейшим 4 моделям

с одним скачком на границе ядро-мантия.

1 Introduction. Determination of the Earth's volume density distribution S(p, ■&, l) from external

potential data requires a solution of the known inverse problem of the Newtonian potential. If the planet's gravitational potential energy E and density at the surface are accepted as additional information, this problem transforms from an improperly posed to a properly posed problem with its possible solution for the 3D density S (p l) through the three-dimensional Cartesian moments (Mescheryakov, 1977). According to Gauss (1840) the search of the stationary value E can be treated as one of central subjects of the potential theory. A remarkable summary of the Gauss' problem reads: "minimum and maximum potential energy correspond to physically (for the Earth) meaningless cases: a surface distribution and a mass point. The 'true' Earth lies somewhere in between" (Moritz, 1990). It is obvious that the potential energy E can be estimated from the density and internal gravitational potential. However only few E-values for the homogeneous Earth (Mescheryakov, 1973; Rubincam, 1979; Moritz, 1990) and the planet differentiated into homogeneous mantle and homogeneous core (Rubincam, 1979) are found in the literature. Thus, the question remains: how can we evaluate better this 'true' Earth and the corresponding potential energy E.

This study focuses on (a) the determination of the Earth's global density distribution and (b) the estimation of the gravitational potential energy E using continuous and piecewise density models. The Earth's mass and principal moments of inertia represent initial information for the unique solution of the restricted Cartesian moments problem providing in this way the density S(p,l) and the potential energy E. The principal moments of inertia given in (Marchenko, 2007) were used for the computation of the 3D global density S(p,l).

It should be pointed out, that accuracy of the global density and potential energy was derived especially to restrict the possible solution domain in such a way that a reasonable solution may be selected either from 3D-spatial or radial density inside the ellipsoidal or spherical planet.

2. The Earth's global density distribution. Let us consider the mathematical model of the 3D global density distribution S(pA) derived by (Mescheryakov et al., 1977) inside the Earth having a shape of the ellipsoid of revolution with the flattening f and the semimajor axis a. According to Mescheryakov (1991) the exact but restricted by the order 2 solution of the three-dimension Cartesian moments problem for S (p   , A) reads

S (p   A) = S (p )r + AS (p   A), (1) AS (p    A) = AK + p 2(AK1sin2 & cos2 A + AK2sin2 & sin2 + AK3cos2 &) , (2) where S (p)R is the piecewise reference radial density model with radial density jumps such as PREM (Dziewonski and Anderson, 1981), AS(pA) is some anomalous density with the following components

AK = 5 S m [5AI

4

000

AI0

AK1 =

35 4

35

sr (3ai200 + AI020 +

AI0

X

AI

AK 2 = =4 S m (AI

200 + 3AI020 +

X AI

000 ),

002

AK3 =

35

sr (ai200 + ai020 + 3

X

AI0

X

AI000),

AI000),

AI000 = AI020 =

R

000 , R, 020 ,

AI200 = AI002 =

R 1

200

R I 002 • I (3)

In the relationships above X=1-f the dimensionless Cartesian moments I 000

I 200

I020 , and I002

I020 = I002 =

(4)

( I100 = I010 = I001 = 0 ) of the density of a gravitating body (Grafarend et al., 2000) can be computed via the Earth's mass Mand dimensionless principal moments of inertia A, B, and C normalized by 1/Ma2:

1000 = 1 ^

1= (B + C - A) 12, (A - B + C)/2, (A + B - C)/2,

The reference model S(p)R includes individual information about density jumps, the mean density Sm , and the mean moment of inertia I1R , which have been selected preliminary for the construction of the radial profile S(p)R . In contrast to Mescheryakov (1991) [Eqs. (1 - 3)] the Cartesian moments 1000 , IR

A B

C

V5A20(1 -1/ hd ) -V15A22/3,1

V5A20(1 -1/ Hd ) + V15A22/3

-V5A20 / H

200

I

020 and I0R02 of the reference density S(p)R were derived here for one common set of the conventional

constants SR and IR of the model (1) and density jumps entering into S(p)R

IR =

000

IR =

200

RR

R  = 3IRS

7020 - „ c ^.2

IR =

002

2S (X2 +2)

3 ■ x2 irrSR

(5)

2 Sr (X2 + 2)'

Thus, in these formulae p (0 £ p £ 1) is the relative distance from the origin of a coordinate system to an internal current point; & and A are the polar distance and longitude of this point; SR is the convenient mean density; Hd is the dynamical ellipticity; A20, A22 are the fully normalized (non-zero) harmonic coefficients adopted here as Stokes constants in the principal axes system OABC . Therefore, this 3D

2

000

200

020

002

R

R

m

R

S

m

m

m

global density [Eq.(1)] is given in the geocentric coordinate system of the principal axes of inertia and agreed with the Earth's mass and the principal moments of inertia to preserve in this way the external gravitational potential from zero to second degree/order, HD, the flattening f and density jumps.

The radial density S(p)R is also treated within the ellipsoid of revolution if we use the formula

re = R(1 - 2f P2(cosJ)/3) for the radius vector re by neglecting f2 (Moritz, 1990), where P2(cos J) is the 2nd-degree Legendre polynomial. This formula results from the average of re over the unite sphere that gives the mean radius R=6371 km. According to Mescheryakov (1991) Eqs. (1 - 2) are valid for a homothetic stratification when f=const inside the ellipsoidal Earth. Hence, if the set of the internal ellipsoidal surfaces ~ is labeled by the associated mean radius r of a sphere we have

r = r

1

2

f P

(cos J)

=> p

У    -      'j      *    R re By averaging S(p,J, l) over ellipsoidal surfaces we define the piecewise radial density S(p) as

[dk + p2 ad\

AI

AD-

35 _ S

12

S(p)=S(p)R

+

L

AI200 + AI020 + "

'002

Ґ

3AI0

jj

(6)

(7)

with the treatment of the reference density S(p)R within the ellipsoidal Earth. Since the radius p is constant for each ~, the densities S(p)R and S(p) are also constant by Eqs. (7) at the surface (6).

Accuracy estimation of the 3D continuous global density was derived from error propagation, keeping in mind that information about accuracy of the mean density STR , the mean moment of inertia ITR , and density jumps in different piecewise radial models S (p )R (such as PREM) are not found in literature or were not easily accessible to the author. For this reason we will consider the reference density S(p)R as some exact constituent or "normal density". Hence, the variance-covariance matrix of the principal moments of inertia, accuracy of the mean density o5 , and accuracy of the flattening sf were chosen as initial information (Table 1).

The accuracy sS of the mean density Sm requires additional remarks because this value represents a scale factor of the considered theory. If Sm = 5.514 g/cm3 and the gravitational constant G = (6.673 ± 0.010) ■ 10-11 m3kg1s 2 suggested by the IERS Conventions 2003 (McCarthy and Petit, 2004)

are selected, we get sS another mean density Sm -- 0.08 g/cm3. According to the IAG recommendations for G and GM (Table 1) = (5.5145 ± 0.0026) g/cm3 finally was adopted.

Table 1

Initial parameters and their accuracy

Reference

Adopted parameters

Groten, 2004

Groten, 2004

Marchenko, 2007

Marchenko, 2007

Marchenko, 2007

Marchenko and Schwintzer, 2003

G=(6.67259±0.0003>10-11 m3kg-1s-2 GM=(398600.4415±0.0008>109m3s"2 v4=0.3296127±0.0000005 5=0.3296200±0.0000005 C=0.3306990±0.0000005 1/f=298.25650 ± 0.00001

Sm =

Thus, the global density distribution and accuracy at different depths were based on the value (5.5145 ± 0.0026) g/cm3, the flatteningf and the principal moments of inertia A, B, and C from Table 1. The principal moments of inertia (given here in the zero frequency tide system) are results from the adjustment of the 2nd-degree harmonic coefficients of 6 gravity field models and 7 values HD of the dynamical ellipticity all transformed to the common value of precession constant at epoch J2000. The

reference radial density profile 5 (p )R in Eq. (1) was selected in the form of the simple piecewise Roche's law separated into seven basic shells (Marchenko, 2000), which is slightly different from PREM.

Fig. 1. Density anomalies [g/cm3] D5 (p ,J, 1) [Eq.(2)]      Fig. 2. Accuracy О 8 (pJ1) [g/cm3] ofthe continuous 3D atthe mantle/crust boundary (r=6346.6 km) density distribution at the mantle/crust boundary

Therefore, with 8(p)R known as exact constituent, the accuracy estimation os(pJ1) of the 3D

continuous global density distribution 8 (p,J,1) (based only on the Earth's mechanical parameters) and lateral density heterogeneities [Eq. (2)] are straightforward. Comparison of these lateral density anomalies D8(p,J,1) (Fig. 1) with the accuracy o5(pJ1) of the continuous constituent at the same depths (Fig. 2)

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