EXPANDING EARTH HYPOTHESIS AND THE EARTH'S GRAVITATIONAL POTENTIAL ENERGY D e d i c a t e d to the M e m o r y o f K. P ~ MILANBtJR~A

Astronomical Institute, Acad. Sci. Czech Republic, Prague 1 ONDI~EJKAHOVORKOVA

University o f N e w Brunswick, Canada

S u m m a r y : The gravitational potential energy o f the actual Earth has now been estimated at - 2 . 4 8 5 x 1032 J. Three density models o f the hypothetical Earth be£ore the hypothetical expansion have been adopted for estimating the internal energy necessary for expanding: 7 x 103 1J. N o dynamical evidence exists for the origin o f this energy in the last 4 5 0 x 106)I. The hypothetical increase in the Earth's mass M o f about 0.4 M since 2 x 10s y B.P. required for a dynamical balance has also no support in the paleodynamics o f the Earth-Moon-Sun system since the M. Ordovician. 1. INTRODUCTION In many papers dealing with the expanding Earth hypothesis, summarized, e.g., in the recent book by Carey, it is believed that the Earth's radius was significantly smaller in the past, "before expanding" (BE), as compared to its actual value R. It is believed, e.g., that at the time of BE assumed to be about 2 x 108 y before the present (B.P.), RBE = 0.6 R (Carey [1]). If this were true, the origin of the energy responsible for this dramatic expansion should be defined. That is why we wish to determine the gravitational potential energy U of the Earth's body at present and UBE at the epoch of BE. The difference

AU= U - UBE

(1)

should be discussed. 2. GRAVITATIONAL POTENTIAL ENERGY OF THE ACTUAL EARTH

The density model given in [2] consisting of 93 homogeneous spherical layers (i = 1, 2 ..... 93) will be adopted. The internal gravitational potential Vint required to determine the gravitational potential energy U of the model

u=-l II dmdm'=-7-MM 1

Address:

fjVintdm

(2)

M

Bo6nflI/1401, 141 31 Praha 4, Czech Republic

Stadia geoph, et geod. 38 (1994)

235

M. Bur~a and O. Hovorkov~

can be expressed as k-I

Vint(P) = 4 ~ G 1 Z Gi(R3- R3 1) + 2gGGkR~- 2~GcrkP 2 -

J

Pi=t

- -3rcocrk

~

P

-

;

+ 21¢Gi= k+ 1

AVint(P) = - 41¢G{rk

(3)

(4)

Notations: G is the Newtonian gravitational constant, M the total Earth's mass, dm and din' its elements separated by distance r is the geocentric radius-vector of the internal point assumed to be situated within the k-th layer, Ri-i,Ri (Rk-t,Rk), are spherical radii defining the i-th (k-th) layer, o'i and o'k are the corresponding densities. After inserting (3) into (2) and integrating over the i-th layer, the contribution AU~ of this layer to the total gravitational potential energy (2) is as follows

~e0; p

av,=

-

-~/t42o , --215~5.+ 6.,R~. 1-2R~'-IR/2+ cr~.l(R/2 R/2. 1)x r, -

i-1 X j=l

93

x

~

aj

j- I

)]

(5)

j= i+ 1

The numerical values are summarized in Table 1. The total gravitational potential energy of the Earth comes out as U = -2.4849 x 1032 J

(6)

The contribution of the mantle to the total value above is about 61%, of the lower mantle 46.5%, of the upper mantle 14.5%, and of the core about 38.6% (Table 1). The spin angular momentum of the i-th layer corresponding to (5)

ALl = o~ACi

(7)

is summarized in Table 2. The total Earth's spin angular momentum comes out as L = 5.8652 × 1033 k g m 2s -1

236

,

(8)

Studia geoph, et geod. 38 (1994)

Expanding Earth Hypothesis and the Earth's... Table 1. Distribution of the gravitational potential energy of the Earth. Body

Radii [km]

Numbers of layers

-AU [1031kgm2s -2]

-Y~AU [1031kgm2s-2]

AU/U [%]

Inner Core Outer Core Core

0-1215 1215-3485 0-3485

1-12 13-35 1-35

0.515 9-209 9.724

0.515 9.724 -

2.1 36.5 38.6

Lower mantle Upper m. a) b) Upper mantle Mantle

3485-5700 5700-5951 5951-6350 5700-6350 3485-6350

36-67 68-77 78-91 68-91 36-91

11.522 1.395 2.104 3.499 15-021

21.246 22.641 24.745 -

46.5 5.9 8.6 14.5 61-0

Crust Earth

6350-6371 O--6371

91-93 1-93

0.104 24.849

24-849 24.849

0.4 100-0

the contribution of the mantle amounts about 87-5% (of the lower mantle 55-4%, of the upper mantle 32.1%), and that of the core about 11.5%. The angular velocity in (7) was adopted constant for all the layers: o~--7.292115 x 10-5rads -l ; ACj in (7) denotes the principal moment of inertia of the i-th layer, V|nt the internal gravitational potential (3) at the lower boundary sphere. The radius of inertia of the actual Earth is Rlneaia = (CIM) 1/2 = 3668 km

;

(9)

C = 8.0358 x 1037 kg m e is the maximum principal moment of inertia of the Earth, and M its total mass. Ratio Rinertia/R = 0.579

(10)

reflects the concentration of the Earth's mass at its center. Note that for a homogeneous body Rineaia/R = 0-632 ,

(11)

and that value (9) is approximately equal to the radius of the 39th zone (or = 5460 kg m -3) situated in the lower mantle in the model used [2]. Radius Ru defined as GM2/Ru = - U

Stadia g~oph,et geod.3S (Z994)

(12)

237

M. Bur~a and O. Hovorkov~ Table 2. Distribution of the Earth's spin angular momentum. Body

AL [1032kgm2s-1]

Inner Core Outer Core Core

ZAL [1032kgm2s -1]

AL/L [%]

]lint [103 m2s -2]

0-040 6-676 6-716

0.040 6-716 -

11.4 11.5

Lower mantle Upper m. a) b) Upper mantle Mantle

32-471 6.942 11-935 18-877 51.348

39-187 46.129 58-064 -

55-4 11.8 20-3 32.1 87.5

69385 66870 62908 -

Crust Earth

0-588 58.652

58-652 58-652

1.0 100.0

62636 62636

0.1

109150 91752 -

-

comes out as

R u = 9582 km and

Ru/R = 1.504

,

(13)

Rinertia]RU = 0.383

Note that for the homogeneous model RuIR = 1-667, and (14)

Rinertia/R U = 0.379 3. GRAVITATIONAL POTENTIAL ENERGY OF THE EARTH BEFORE ITS HYPOTHETICAL EXPANDING (BE).

Three density models will be dealt with: 1. R o c h e ' s model: the heavy core plus the mantle, the mass of which is practically negligible (the mass concentrated at the centre). 2: N e w t o n ' s model: the homogeneous body. 3. Model consisting of 93 spherical homogeneous layers analogous to that of the actual Earth [2], the radius o f the boundary sphere being RBE = ~ R. Mass MaE of all the models is assumed to be equal to mass M of the actual Earth, i.e. MBE = M = 5.9737 × 1024 kg

,

(15)

because there is no evidence and/or no reason for admitting any dynamically significant variation in M in the last 5 x 10 s years. In that case

238

Stadia geoph, et geod. 38 (1994)

Expanding Earth Hypothesis and the Earth's... M RB2E = 8.7289 X 1037 kg m 2

(16)

To be realistic, hydrostatic equilibrium limits will be respected. Model

I.

Because of zero polar flattening of the concentrated mass, the second zonal Stokes parameter is equal to zero: (J2(°))i= 0 ,

(17)

which also applies to, the secular Love number of model (I)

t/°)l (ks)i=_

3~ 2 / B E = 0 qBE

;

2 3 COBERBE qBE =

(18)

GM

The geocentric gravitational constant adopted is

GM= 398600.441 x 109 m 3 s-2

(19)

Angular velocity tOnE at epoch BE can be estimated on the basis of dco/dt = - (5.0 ± 0.2) x 10-22 rad s-2

(20)

(Rep. IAG. SSG 5 - 100 [3]), see Table 3. At epoch BE assumed to be 2 x 10 s y B.P. it comes out as rOBE= 7.60769 x 10- 5 rad s-1 ,

(21)

and (about 382 days in the tropical year, at epoch BE), and qBE = 8-1104 x 10-4.

(22)

The principal moment of inertia of model (I) comes out as

2 2 = -~MR~E= 1.164 x 1037kg m 2

,

(23)

from which the radius of the core (Rcore)BE can be estimated as ( 5 C I ~ 1/2

(Rc~e)I = ~,~ ~ ' J

Studia geoph, et geod. 38 (1994)

= 2206.98 kin

(24)

239

M. Bu~a and 0. Hovorkov~ Table 3. Angular velocity of the Earth's rotation at geological epochs for which paleontological evidence is available.

Epoch

Present Pleistocene U. Cretaceous U. Permian M. Silurian M. Ordovician

Time [106 B.P.]

oJ [10-5 tad s"1]

[s]

0 0.2 70 240 420 450

7.292115 7.292456 7.411403 7.701101 8.007841 8.058965

86164 86160 84777 81588 78463 77965

T [hi

[m]

Is]

23 23 23 22 21 21

56 56 32 39 47 39

04 00 57 48 43 25

Length of year [d] 365.25 365.27 37! 386 401 404

The mostly concentrated mass is supposed to be homogeneous. The gravitational potential energy of model (I) comes out as UI = -6-473 x 1032 kg m 2 s-2

(25)

The spin angular momentum of the model is

(26)

LI = CI OJBE = 8.855 X 103 2 kg m 2 s "1

and the kinetic energy due to the rotation EI = ½Ci ~02E = 3.368 x 102s kg m 2 s-2

(27)

The radius of inertia (Rlnertla)I = 1396 k m ,

(Rinertla)llRBE = 0.365

(Ru)I= 3679 k m ,

(Ru)I/RsE= 0.962,

(28)

and

Model

(RinenJa)11(Ru)I = 0.379. (30)

II.

The secular Love number is

(31)

(k,)n = 3/2 , the second zonal Stokes parameter

(32) the maximum principal moment of inertia Cn = ~

240

MR2E=

3.492 x 1037 kg m 2 ,

(33)

Studia geoph, et geod. 38 (1994)

F_.xpandinBEarthHypothesisand the Earth's... the spin angular momentum LI]= CnoJnB= 2.657× 1033kgm2s -1

(34)

,

and the kinetic energy due to rotation 1 2 Eli = ~CII¢OBE= 1.011× 1029 kgm2s - 2

(35)

The gravitational potential energy of the model comes out as Un = - 3.737 x 1032 kgm2s - 2 ,

(36)

the radius of ine1~ia (Rinmla)ii = 2418km

,

(Rinertia)ii/RBE = 0" 632

(37)

(Ru)u/RB~= 1-667

(38)

and

(Ru)II 6372 km ----

,

,

(39)

(Rin~ia)U/(Ru)U = O. 379 Model III.

Because of the similarity with the actual Earth as regards internal structure, i.e. the relative distribution of densities being the same, only the radii of the spheres separating the 93 layers differ: =

(4O)

In that case Urn= ~ U = - 4.1415x 1032kgm2s - 2 ,

(41)

CIII = 2 . 8 8 6 × 1037 kgm 2 ,

(42)

and

Cm

- 0.331

M R2E LIII= CIIIOJBE= 2. 196× 1033 kgm2s -1

Studia geoph, et geod. 38 (1994)

(43)

24 1

M. BurJa

and O. Hovorkovd

1 EIII = ~CIIIOj2E = 8 . 3 5 8 × 102s k g m 2 s - 2

(44)

As regards (ks)Ill and (J2(°))lli, the limits can be estimated adopting (23): (45)

--(./(20))Iii= 0"000270(ks)Ii I On the other hand, assuming hydrostatic equilibrium,

=

-

0.000252

(46)

and using (45) (47)

(ks)ill = 0"933

The radius of inertia (Rinertia)iii = 2198km

,

(Rinetlia)lll/RBE=0"575 ,

(48)

(RU)III/RBE=1"504

(49)

and

(Ru)III = 5 7 4 9 k m

,

(Rinertia)III/(RU)III = 0"382

,

(50)

4. DIFFERENCES BETWEEN GEODYNAMICAL PARAMETERS AT PRESENT AND BE Differences (1), as well as AL=L-LBE, ARlnertia=ginertia - (Rinertia)BE, ARu= =Ru-(Rv)BE, and AE=E-EBE for all three models are summarized in Tables 4 and 5. The total mechanical energy of the Earth's body

H= U+E

(51)

should change dramatically if the expanding Earth hypothesis were realistic. The actual value (at presen0 is H = - 2. 4846 x 1032 J (52) and at epoch BE (models I, II, III) H I=-

6- 4 7 3 x 1 0 3 2 J

HII = - 3. 736 x 1032 J

, ,

HIi I = - 4. 141 x 1032 J . 242

(53) Studia geoph, et geod. 38 (1994)

Expanding Earth Hypothesis and the Earth's ... The minimum energy required for expansion comes out as (model II)

A H = + 1.25 x 1032 J

(54)

If the expanding were linear in time, starting at epoch 2 x l0 s y B.P., then

d H / d t = 1.98 x 1016 W

(55)

Note that the actual variations in H, observed at present, of tidal and nontidal origin are about four orders of magnitude smaller:

(rill/dr)tidal= - 3 . 7 1 x 1012 W

,

(56)

(dH/dt)aontidal = + 0 . 3 8 x 1012W

(57)

The spin angular momentum of the Earth should also have increased dramatically, if the Earth's expanding hypothesis were true. The actual value (at present) is (8), and the minimum variation (model II) comes out as

A L = + 3-21 x 1033 k g m 2 s -1 ,

(58)

dL/dt=

(59)

+ 5 . 0 4 x 1017kgm2s - 2

Table 4. Differences of radii of inertia at present and BE, secular Love numbers and Stokes zonal parameters.

C/MR2E

ks

Model

I H III

0 3/2 0.933

2/15 2/5 0.331

-./2 (0)

ARinerti a

[10--61

[kin]

0 406 252

2272 1250 1470

A Rinertia/Rine~ a

0.619 0.341 0-401

Table 5. Differences between gravitational potential energy, spin angular momentum and kinetic energy due to rotation at present and BE. Model

AU

AU U

[1032 J] I II IlI

3.988 1.252 1.657

AL

__AL L

[1033kgm2s-l] 1.60 0.50 0.67

Stadia geoph, et geod. 38 (1994)

4.980 3.208 3.669

AE

AE

[1029 J] 0.85 0.55 0.63

ARu

E

1.800 1.126 1.301

AR U RU

[kin] 0-84 0-53 0.61

5903 3210 3833

0.616 0.335 0-400 243

M. Bur~a and O. Hovorkovti

For comparison, the actual (observed) long-term variations in L due to tidal friction and of nontidal origin are about (dL/dt)tidal,~ - 5 x 1016kgm2s - 2 (dL/dt)aoatids1- + I x 10t6kgm2s - 2

(60)

Variations (54), (55) and (58), (59) were derived assuming MI3E= M = the actual Earth's mass at present. The dramatic variation in M necessary for the dynamical compensation of the expansion and its absolutely nonrealistic basis has been discussed in [4]. If no other source for explaining the increase in energy (54), and putting UBE = U, admitting Rnp.=0.6R, A R = - O . 4 R , the increase in Earth's mass required for compensation (55) then comes out as AM _ M

1 AR = _2 2R 5

(61)

In other words, the absolutely nonrealistic increase of 0.4 M should have occured. Nothing similar has been detected in the Earth-Moon-Sun paleodynamics during the period of about - 450 x 106 y, i.e. since the M. Ordovician. Paleontologic data [5] is weighing evidence in favour of this statement. The secular increase in the semimajor axis of the Moon's orbit and/or the secular decrease in its mean motion is well explained by tidal friction [6]. The angular velocity of the Earth's rotation has decreased during the last 500 × 106 y by about 10%. The tidal friction dynamics of the Earth-Moon-Sun system calls for a value about 0.2 larger, and there is absolutely no scope at all for assuming that the angular velocity has decreased due to the hypothetical expanding. 5. CONCLUSIONS 1. The gravitational potential energy of the actual Earth (at presen0 has been estimated as U = - 2 . 4 8 5 × 1032 J. 2. Since the actual Earth's body is relatively close to hydrostatic equilibrium, Poincar~'s virial theorem [7] gives an estimate of the internal energy of the Earth (at present) of a b o u t - 1.24 x 1032 J. 3. The internal energy of the Earth required for expansion is about - 7 x 1031 J. No realistic origin of such energy could be found "in the Earth-Moon-Sun system dynamics in the last - 450 x 106 y. Received: 6. 9. 1993 Revised: 13. 1. 1994

244

Studiageoph,et geod.38 (1994)

Expanding Earth Hypothesis and the Earth's... References [1] S. W. C a r e y : Theories of the Earth and Universe. Stanford University Press, Stanford, California, 1988. [2] J. A. J a c o b s : The Earth's Core. Academic Press Inc., London, New York, San Francisco 1975. [3] R e p o r t o f I A G SSG 5-100: Parameters of Common Relevance of Astronomy, Geodesy, and Geodynamics. Presented at XXth IUGG/IAG Gen. Ass., Vienna 1991, Bull. G63d., 66 (1992), 193-197. [4] M. Bur]~a: Global geodynamic long-term variations and expanding Earth hypothesis. Studia geoph, et geod., 37 (1993), 113-124. [5] C. T. S c r u t t o n : Periodic growth features in fossil organtsmus and the length of the day and Month. In: P. Brosche, J. SUndermann (Eds.), Tidal Friction and the Earth's Rotation, Springer-Verlag, Berlin, Heidelberg, New York 1978, 154--196. [6] F i n a l R e p o r t SSG 5-99: Tidal Friction and the Earth's Rotation 1983-1987. In: C. Boucher (Ed.), Travaux de rAssociation Internationale de G6od6sie, 28, Paris 1988, 491-513. [7] H. P o i n c a r 6 : Lecons sur les Hypotheses Cosmogoniques, Librairie Scientifique A. Hermann et Fils, Paris, 1913, pp. 294 (in French).

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